Calculus primer

8/7/2019 Calculus primer

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Increasing and Decreasing FunctionsIncreasing Functions

A function is "increasing" if the y-value increases as the x-value increases, like this:

It is easy to see that y=f(x) tends to go up as it goes along .

Flat ?What about that flat bit near the start? Is that OK?• Yes, it is OK if you say the function is Increasing

• But it is not OK if you say the function is Strictly Increasing (no flatness allowed)

Using Algebra

What if you can't plot the graph to see if it is increasing? In that case is is good to have adefinition using algebra.

For a function y=f(x) :

when x 1< x 2 then f(x 1) ≤ f(x 2) Increasingwhen x 1< x 2 then f(x 1) < f(x 2) Strictly Increasing

That has to be true for any x1, x 2, not just some nice ones you choose.

The important parts are the < and ≤ signs ... remember where they go!

An Example:

http://www.mathsisfun.com/sets/function.html

http://www.mathsisfun.com/equal-less-greater.html



http://www.mathsisfun.com/sets/function.html



This is also an increasing functioneven though the rate of increase reduces

For An IntervalUsually you will only be interested in some interval , like this one:

This function is increasing for the interval shown(it may be increasing or decreasing elsewhere)

Decreasing FunctionsThe y-value decreases as the x-value increases:



For a function y=f(x) :

when x 1< x 2 then f(x 1) ≥ f(x 2) Decreasing

when x 1< x 2 then f(x 1) > f(x 2) Strictly DecreasingNotice that f(x 1) is now larger than (or equal to) f(x 2).

An ExampleLet us try to find where a function is increasing or decreasing

Example: f(x) = x 3-4x, for x in the interval [-1,2]Let us plot it, including the interval [-1,2]:

Starting from -1 (the beginning of the interval [-1,2] ):• at x = -1 the function is decreasing,• it continues to decrease until about 1.2• it then increases from there, past x = 2

Without exact analysis we cannot pinpoint where the curve turns from decreasing to increasing,so let us just say:

Within the interval [-1,2] :• the curve decreases in the interval [-1, approx 1.2]

• the curve increases in the interval [approx 1.2, 2]

Constant FunctionsA Constant Function is a horizontal line:



LinesIn fact lines are either increasing, decreasing, or constant.

The equation of a line is:

y = mx + bThe slope m tells us if the function is increasing, decreasing or constant:

m < 0 decreasingm = 0 constantm > 0 increasing

One-to-OneStrictly Increasing (and Strictly Decreasing) functions have a special property called "injective"

or "one-to-one" which simply means you never get the same "y" value twice.

General Function "Injective" (one-to-one)



Why is this useful? Because Injective Functions can be reversed !

You can go from a "y" value back to an "x" value (which you couldn't do if there were morethan one possible "x" value).

Read Injective, Surjective and Bijective to find out more.

Maxima and Minima of FunctionsLocal Maximum and Minimum

Functions can have "hills and valleys": places where they reach a minimum or maximum value.

It may not be the minimum or maximum for the whole function , but locally it is.

http://www.mathsisfun.com/sets/injective-surjective-bijective.html





You can see where they are,but how do we define them?

Local MaximumFirst we need to choose an interval:

Then we can say that a local maximum is the point where:

The height of the function at "a" is greater than (or equal to) the height anywhere else in thatinterval.

Or, more briefly:

f(a) ≥ f(x) for all x in the interval

In other words, there is no height greater than f(a).

Note: f(a) should be inside the interval, not at one end or the other.

Local MinimumLikewise, a local minimum is:

f(a) ≤ f(x) for all x in the interval

The plural of Maximum is Maxima

The plural of Minimum is Minima



Maxima and Minima are collectively called Extrema

Global (or Absolute) Maximum and MinimumThe maximum or minimum over the entire function is called an "Absolute" or "Global"

maximum or minimum.

There is only one global maximum (and one global minimum) but there can be more than onelocal maximum or minimum.

Assuming this function continuesdownwards to left and right:

• The Global Maximum is about3.7

• The Global Minimum is

-Infinity

CalculusCalculus can be used to find the exact maximum and minimum values of many functions, but

that is beyond this article.



Derivatives as dy/dxDerivatives are all about change ...... they show how fast something is changing (called the rate of change ) at any point.

In Introduction to Derivatives (please read it first!) we looked at how to do aderivative using differences and limits .

Here we look at doing the same thing but using the "dy/dx" notation (alsocalled Leibniz's notation ) instead of limits.

We start by calling the function "y":y = f(x)

1. Add ΔxIf x increases by Δx, then y increases by Δy

y + Δy = f(x + Δx)

2. Subtract the Two FormulasFrom: y + Δy = f(x + Δx)

Subtract: y = f(x)

To Get: y + Δy - y = f(x + Δx) - f(x)

http://www.mathsisfun.com/calculus/derivatives-introduction.html

http://www.mathsisfun.com/calculus/derivatives-introduction.html



Simplify: Δy = f(x + Δx) - f(x)

3. Rate of ChangeTo work out how fast (called the rate of change ) we divide by Δx :

4. Reduce Δx close to 0We can't let Δx become 0 (because that would be dividing by 0), but we canmake it very small , and call it "dx":

Δx dx

You can think of "dx" as being infinitesimal , or infinitely small.

Likewise Δy becomes very small and we call it "dy", to give us:

Try It On A FunctionLet's try f(x) = x 2

f(x) = x 2



Expand (x+dx) 2

Simplify (x 2-x 2=0)

Simplify fraction

dx goes (infinitely small)



How Polynomials BehaveA polynomial looks like this:

example of a polynomial

Continuous and SmoothThere are two main things about the graphs of Polynomials:

The graphs of polynomials are continuous , which is a special term with an exact definition incalculus, but here we will use this simplified definition:

you can draw it without lifting your pen from the paper

The graphs of polynomials are also smooth . No sharp "corners" or "cusps"

How the Curves BehaveLet us graph some polynomials to see what happens ...

... and let us start with the simplest form:

f(x) = x n

Which actually does interesting things:

http://www.mathsisfun.com/algebra/polynomials.html

http://www.mathsisfun.com/algebra/polynomials.html



Even values of "n" behave the same:• Always above (or equal to) 0• Always go through (0,0), (1,1) and (-1,1)• Larger values of n flatten out near 0, and rise more

sharply

And:

Odd values of "n" behave the same• Always go from negative x and y to positive x and y• Always go through (0,0), (1,1) and (-1,-1)• Larger values of n flatten out near 0, and fall/rise

more sharply

Power Function of Degree nNext, by including a multiplier of a we get what is called a "Power Function":

f(x) = ax n

f(x) equals a times x to the "power" (ie exponent) nThe "a" changes it this way:

• Larger values of a squash the curve (inwards to y-axis)• Smaller values of a expand it (away from y-axis)

• And negative values of a flip it upside down

Example: f(x) = ax 2

a = 2, 1, ½, -1Example: f(x) = ax 3

a = 2, 1, ½, -1



We can use that knowledge when sketching some polynomials:

Example: Make a Sketch of y=1-2x 7

Start with the simplest "odd power" graph of x 3, and gradually turn it into 1-2x 7

• You know how x 3 looks,• x7 will be similar, but flatter near zero, and steeper elsewhere,

• Squash it to get 2x 7,• Flip it to get -2x 7, and

• Raise it by 1 to get 1-2x 7.

Like this:

So by doing this step-by-step we can get a good result.

Turning PointsA Turning Point is an x-value where a local maximum or local minimum happens:

http://www.mathsisfun.com/algebra/functions-maxima-minima.html





How many turning points does a polynomial have?Never more than the Degree minus 1

The Degree of a Polynomial with one variable is the largest exponent of that variable.

Example: a polynomial of Degree 4 will have 3 turning points or less

x4-2x 2+xhas 3 turning points x4-2x

has only 1 turning point

The most is 3, but there can be less.

You may not know where they are, but at least you know the most there can be!

What Happens at the EndsAnd when you move far from zero:

• far to the right (large values of x), or

http://www.mathsisfun.com/algebra/degree-expression.html



http://www.mathsisfun.com/exponent.html





• far to the left (large negative values of x)

then the graph starts to resemble the graph of y = ax n where ax n is the term with the highestdegree .

Example: f(x) = 3x 3-4x 2+x

Far to the left or right, the graph will look like 3x3

Near Zero, they aredifferent Far From Zero, they

become similar

This makes sense, because when x is large, then x 3 is much greater than x 2etc

This is officially called the " End Behavior Model ".

And yes, we have come to the end!

Summary• Graphs will be continuous and smooth

• Even exponents behave the same: above (or equal to) 0; go through (0,0), (1,1) and (-1,1); larger values of n flatten out near 0, and rise more sharply.

• Odd exponents behave the same: go from negative x and y to positive x and y; gothrough (0,0), (1,1) and (-1,-1); larger values of n flatten out near 0, and fall/rise more

sharply• Factors:

○ Larger values squash the curve (inwards to y-axis)○ Smaller values expand it (away from y-axis)

○ And negative values flip it upside down• Turning points: there will be "Degree-1" or less.

• End Behavior: use the term with the largest exponent

Intermediate Value TheoremThe idea behind the Intermediate Value Theorem is this:



When you have two points connected by a continuous curve:• one point below the line•

the other point above the line... then there will be at least one place where the curvecrosses the line!

Well of course you must cross the line to get from A to B!

Now that you know the idea , let's look more closely at the details.

ContinuousThe curve must be continuous ... no gaps or jumps in it.

"Continuous" is a special term with an exact definition in calculus, but here we will use thissimplified definition:

you can draw it without lifting your pen from the paper

More FormalHere is that idea stated more formally:

When:• The curve is the function y = f(x),• which is continuous on the interval [a, b],• and w is a number between f(a) and f(b),

Then ...

... there must be at least one value c within [a, b] such that f(c) = w

In other words the function y = f(x) at some point must be w = f(c)

Notice that:• w is between f(a) and f(b), which leads to ...

• c must be between a and b



At Least One

It also says "at least one value c", which means youcould have more.

Here, for example, are 3 points where f(x)=w.

How Is This Useful?Whenever you can show that:

• there is a point above a line

• and a point below a line, and• that the curve is continuous,

you can then safely say "yes, there is a value somewhere in between that is on the line".

Example: is there a solution to x 5 - 2x 3 - 2 = 0 between x=0 and x=2?At x=0:

05 - 2 × 0 3 - 2 = -2

At x=2:

25 - 2 × 2 3 - 2 = 14

Now we know:• at x=0, the curve is below zero• at x=2, the curve is above zero

And, being a polynomial, the curve will be continuous,

so somewhere in between, the curve must cross through y=0

Yes, there is a solution to x 5 - 2x 3 - 2 = 0 in the interval [0, 2]

An Interesting Thing!The Intermediate Value Theorem Can Fix a Wobbly Table





two points that aredirectly opposite and at same height

DerivativesDerivatives are all about change ...... they show how fast something is changing (called the rate of change ) at any point.

Let us get straight into an example: the function x 2 :f(x) = x 2

Here are some values:

x f(x)

0 0

0.5 0.25

1 1

1.5 2.25

2 4

2.5 6.25



As x increases, so does f(x) ... but how fast ?

Can we find the slope (or rate of change ) atany point?With Derivatives we can!

DifferencesFirst, let's calculate some differences.

The symbol for difference is the greek letter "delta":"Δ"

Example: if t goes from 3 to 3.1, then Δt =0.1

Here are some differences for x and f(x)

x Δx f(x) Δf(x)

0 0

0.5 0.5 0.25 0.25

1 0.5 1 0.75

1.5 0.5 2.25 1.25

2 0.5 4 1.75

2.5 0.5 6.25 2.25

The table shows that, for example, between 1.5 and 2 f(x) grew by 1.75

I kept Δx at 0.5, but Δf(x) is getting larger and larger, meaning the rate of change is increasing.

But now we introduce the first of our important steps:

1. Make a FormulaInstead of long tables of numbers, we want a formula for calculatingΔf(x) .

What happens between x and x+Δx?



At x: At x+Δx:

f (x) = x 2 f (x+Δx) = (x + Δx) 2

So, the difference is:

Δf(x) = f (x + Δx) - f (x)

= (x + Δx) 2 - x 2

2. Simplify Formula

We can expand "(x + Δx) 2": Δf(x) = x 2 + 2x Δx + Δx 2 - x 2

And then simplify the formula: Δf(x) =

Now we can calculate Δf(x) in one go:

Example: What is Δf(x) for x=1.5 and Δx=0.5 ?

Δf(x) = 2 x Δx + (Δx)2

= 2 × 1.5 × 0.5 + 0.5 × 0.5 = 1.5 + 0.25 = 1.75

That tells us how much f(x) changes, but not how fast . We still need tocompare it to the change in x.

Example: if my weight increased by 1kg, was thatfast or slow ?

You can only answer if you know how long it took toincrease. If it happened in 1 day, then it is fast, but if it took 1 year, it is slow.

3. Rate of ChangeTo work out how fast (called the rate of change ) we divide by Δx :

Rate of change = Δf(x) / Δx

That tells us "how much f(x) changes for every change in x"



The Rate of Change can be seen as the slope of a line:

In our x 2 example, this becomes:

Δf(x) / Δx =Example: What is the rate of change for Δf(x) when x=1.5 andΔx=0.5 ?

Rate of change = Δf(x) / Δx

= ( 2 x Δx + (Δx)² ) / Δx

= (2 × 1.5 × 0.5 + 0.5 2)/0.5

= 1.75/0.5

= 3.5

This tells us the slope of the line between x and(x+Δx).

But Not At A Point

But it doesn't yet tell us the slope at any specificpoint ...

... for example what is the slope at 1.5?

But we can find the slope at a point if we:

Reduce Δx towards 0.

That will show us the slope just where 1.5 is.

However, Δx can't be 0 , because dividing by 0 is wrong! ...

... but we can try getting closer and closer:

Δx = 0.5 : Rate of change (we already calculated) = 3.5

Δx = 0.05 : Rate of change = (2 × 1.5 × 0.05 + 0.05 2)/0.05 = 0.1525/0.05 =3.05



Δx = 0.005 : Rate of change = (2 × 1.5 × 0.005 + 0.005 2)/0.005 = 3.005

We seem to be heading for Rate of change = 3

4. Use Limits to get Δx close to 0In fact we just used the idea of limits to find the Rate of Change!

Because we weren't allowed to have Δx = 0, we tried approaching it closerand closer.

So let's try putting our Rate of Change formula into a limit, and see what weget:

The first thing we can do is simplify it , because Δx is at both the top andbottom:

Now, as Δx goes towards zero we have:

And that is it ... the rate of change (the slope of the line) is simply 2xWe have solved it! We can find the slope of the line at any point.

The slope at 1.5 is 2 × 1.5 = 3, ... the slope at 10 is 2 × 10 = 20, etc ...And even better, we have done our first derivative!

Those Steps AgainSo how did we do this again?

1. We wrote a formula for Δf(x) = f(x+Δx) - f(x)

2. We simplified that formula (to "2x Δx + Δx²")

3. We divided that by Δx to get the rate of change

http://www.mathsisfun.com/calculus/limits.html




4. We used limits to get Δx close to 0.

Definition of a DerivativeLet us now imagine that f(x) is any function , and follow the steps:

1. A formula for Δf(x)Δf(x) = f(x+Δx) - f(x)

2. Simplify(can't simplify in this case)

3. Divide by Δx to get the rate of changeΔf(x)/Δx = (f(x+Δx) - f(x))/Δx

4. Use limits to get Δx close to 0.

And that is the Defintion of a Derivative , for which we often use the littleprime mark (') as shown.

Try It On Another FunctionLet's try x 3

A formula for Δf(x)Δf(x) = f(x+Δx) - f(x) = (x+Δx) 3 - x 3

Simplify(x+Δx) 3 - x 3 = x 3 + 3x 2Δx + 3xΔx 2 + Δx 3 - x 3

= 3x 2Δx + 3xΔx 2 + Δx 3

Divide by Δx to get the rate of changeΔf(x)/Δx = (3x 2Δx + 3xΔx 2 + Δx 3)/Δx

Use limits to get Δx close to 0.

We can simplify it, then work out what happens when Δx goes towards 0:

So the derivative of x 3 is 3x 2

Other Notations



Sometimes the derivative is written like this:

Limits (An Introduction)ApproachingSometimes you can't work something out directly ... but you can see what it should be asyou get closer and closer!

Let's use this function as an example:(x 2-1)/(x-1)

And let's work it out for x=1:

(12

-1)/(1-1) = (1-1)/(1-1) = 0/0

Now 0/0 is a difficulty! We don't really know the value of 0/0, so we needanother way of answering this.

So instead of trying to work it out for x=1 let's try approaching it closerand closer:

x (x 2 -1)/(x-1)

0.5 1.50000

0.9 1.90000

0.99 1.99000

0.999 1.99900

0.9999 1.99990

0.99999 1.99999

... ...

Now we can see that as x gets close to 1, then (x 2-1)/(x-1) gets close to 2

We are now faced with an interesting situation:• When x=1 we don't know the answer (it is indeterminate )

• But we can see that it is going to be 2

We want to give the answer "2" but can't, so instead mathematicians sayexactly what is going on by using the special word "limit"

The limit of (x 2-1)/(x-1) as x approaches 1 is 2



And it is written in symbols as:

So it is a special way of saying, "ignoring what happens when you get there,but as you get closer and closer the answer gets closer and closer to 2"

As a graph it looks like this:

So, in truth, you cannot say what the valueat x=1 is.

But you can say that as you approach 1, thelimit is 2.

Test Both Sides!

It is like running up a hill andthen finding the path ismagically "not there"...

... but if you only check oneside, who knows whathappens?

So you need to test it from

both directions to be surewhere it "should be"!

So, let's try from the other side:

x (x 2 -1)/(x-1)

1.5 2.50000

1.1 2.10000

1.01 2.01000

1.001 2.00100

1.0001 2.00010

1.00001 2.00001

... ...



Also heading for 2, so that's OK

When it is different from different sidesWhat if we have a function "f(x)" with a "break" in it like this:

This is a function where the limitdoes not exist at "a"... !

You can't say what it is ,because there are two competinganswers:

• 3.8 from the left, and

• 1.3 from the right

But you can use the special "-" or"+" signs (as shown) to define one

sided limits:• the left-hand limit (-) is 3.8

• the right-hand limit (+) is 1.3

And the ordinary limit "does notexist"

Are limits only for difficult functions?Limits can be used even if you know the value when you get there !Nobody said they are only for difficult functions.

For example:

We know perfectly well that 10/2 = 5, but limits can still be used (if youwant!)

Approaching Infinity

Infinity is a very special idea. We know we can't reach it, but wecan still try to work out the value of functions that have infinity inthem.

Let's start with an interesting example.

Question: What is the value of 1 / ∞ ?

http://www.mathsisfun.com/numbers/infinity.html






And write it like this:

In other words:As x approaches infinity, then 1/x approaches 0

When you see "limit", think "approaching"

It is a mathematical way of saying "we are not talking about when x= ∞, but we know as x gets bigger, the answer gets closer and closer to 0 " .

Read more at Limits to Infinity .Solving!We have been a little lazy so far, and just said that a limit equals some valuebecause it looked like it was going to .

That is not really good enough!

http://www.mathsisfun.com/calculus/limits-infinity.html






So let's start with the general idea

From English to MathematicsLet's say it in English first:

"f(x) gets close to some limit as x gets close to some value"

If we call the Limit "L", and the value that x gets close to "a" we can say

"f(x) gets close to L as x gets close to a"

Calculating "Close"Now, what is a mathematical way of saying "close" ... could we subtract one value from theother?

Example 1: 4.01 - 4 = 0.01

Example 2: 3.8 - 4 = -0.2

Hmmm ... negatively close? That doesn't work ... we really need to say "I don't care aboutpositive or negative, I just want to know how far" The solution is to use the absolute value .

"How Close" = |a-b|

Example 1: |4.01-4| = 0.01

Example 2: |3.8-4| = 0.2

And if |a-b| is small we know we are close, so we write:

"|f(x)-L| is small when |x-a| is small"

And this animation shows youwhat happens with the function

f(x) = (x 2 - 1) / (x-1)

• as x approaches a=1,• f(x) approaches L=2

So• |f(x)-2| is small• when |x-1| is small.



Delta and EpsilonBut "small" is still English and not "Mathematical-ish".

Let's choose two values to be smaller than :

that |x-a| must be smaller than

that |f(x)-L| must be smaller than

(Note: Those two greek letters, δ is "delta" and ε is "epsilon", are oftenused for this, leading to the phrase "delta-epsilon")

And we have:

"|f(x)-L|< when |x-a|< "

That actually says it! So if you understand that you understand limits ...

... but to be absolutely precise we need to add these conditions:

1) 2) 3)

it is true for any >0 exists, and is >0x not equal to a means 0<|x-a|

And this is what we get:

"for any >0, there is a >0 so that |f(x)-L|< when 0<|x-a|< "That is the formal definition. It actually looks pretty scary, doesn't it!

But in essence it still says something simple: when x gets close to a then f(x) gets close to L .

How to Use it in a Proof To use this definition in a proof, we want to go

From: To:

0<|x-a|< |f(x)-L|<

This usually means finding a formula for (in terms of ) that works.

How do we find such a formula?

Guess and Test!

That's right, you can:



1. Play around till you find a formula that might work

2. Test to see if that formula works.

Example: Let's try to show that

Using the letters we talked about above:• The value that x approaches, "a", is 3• The Limit "L" is 10

So we want to know:

How do we go from:0<|x-3|<

to|(2x+4)-10|<

Step 1: Play around till you find a formula that might work

Start with:|(2x+4)-10|<

Simplify:|2x-6|<

Move 2 outside:2|x-3|<

Move 2 across:|x-3|< /2

So we can now guess that = /2 might work

Step 2: Test to see if that formula works.

So, can we get from 0<|x-3|< to |(2x+4)-10|< ... ?

Let's see ...

Start with:0<|x-3|<

Replace : 0<|x-3|< /2

Move 2 across:0<2|x-3|<





Limits (Evaluating)You should read Limits (An Introduction) first

Quick Summary of LimitsSometimes you can't work something out directly ... but you can see what it should be asyou get closer and closer!

For example: (x 2 -1)/(x-1)

At x=1: (1 2-1)/(1-1) = (1-1)/(1-1) = 0/0

But 0/0 is "indeterminate", meaning we can't determine its value. Butinstead of trying to work it out for x=1 let's try approaching it closer andcloser:

x (x 2 -1)/(x-1)

0.5 1.50000

0.9 1.90000

0.99 1.99000

0.999 1.99900

0.9999 1.99990







(1-1)/(1-1) = 0/0

10/2 = 5

It didn't work with the first one (we knew that!), but the second examplegave us a quick and easy answer.

2. Factors

You can try factoring.

Example:

By factoring (x 2-1) into (x-1)(x+1) we get:

Now we can just substitiute x=1 to get the limit:

3. Conjugate

If it's a fraction, then multiplying top and bottom by a conjugate might help.

http://www.mathsisfun.com/algebra/conjugate.html

http://www.mathsisfun.com/algebra/conjugate.html



The conjugate is whereyou change the sign in the

middle of 2 terms like this:

Here is an example where it will help you to find a limit:

Evaluating this at x=4 gives 0/0,which is not a good answer!

So, let's try some rearranging:

Multiply top and bottom by the conjugate of the top:

Simplify top using :

Simplify top further:

Eliminate (4-x) from top and bottom:

So, now we have:

Done!



4. Infinite Limits and Rational Functions

A Rational Function is one that is the ratio of two polynomials:

For example, here P(x)=x 3 +2x-1 , and Q(x)=6x 2 :

By finding the overall Degree of the Function we can find out whether thefunction's limit is 0, Infinity, -Infinity, or easily calculated from thecoefficients.

Read more at Limits To Infinity .

5. Formal MethodThe formal method sets about proving that you can get as close as youwant to the answer by making "x" close to "a".

Read more at Limits (Formal Definition)

Limits to InfinityYou should read Limits (An Introduction) first

http://www.mathsisfun.com/algebra/rational-expression.html



http://www.mathsisfun.com/calculus/limits-formal.html





http://www.mathsisfun.com/calculus/limits-formal.html




Infinity is a very special idea. We know we can't reach it, but wecan still try to work out the value of functions that have infinity inthem.

One Divided By InfinityLet's start with an interesting example.

Question: What is the value of 1 / ∞ ?

Answer: We don't know!

Why don't We know?

The simplest reason is that Infinity is not a number, it is an idea. So 1/ ∞ is abit like saying 1/beauty or 1/tall.

Maybe we could say that 1/ ∞ = 0, ... but that is a problem too, because if we divide 1 into infinite pieces and they end up 0 each, what happened tothe 1?

In fact 1/ ∞ is known to be undefined .

But We Can Approach It!So instead of trying to work it out for infinity (because we can't get asensible answer), let's try larger and larger values of x:

x 1/x

1 1.00000

2 0.50000

4 0.25000

10 0.10000

100 0.01000

1,000 0.00100

10,000 0.00010

Now we can see that as x gets larger, 1/x tends towards 0





We are now faced with an interesting situation:• We can't say what happens when x gets to infinity

• But we can see that 1/x is going towards 0

We want to give the answer "0" but can't, so instead mathematicians say

exactly what is going on by using the special word "limit"The limit of 1/x as x approaches Infinity is 0

And write it like this:

In other words:As x approaches infinity, then 1/x approaches 0

When you see "limit", think "approaching"

It is a mathematical way of saying "we are not talking about when x= ∞, but we know as x gets bigger, the answer gets closer and closer to 0 " .

Summary

So, sometimes Infinity cannot be used directly, but you can use a limit.

What happens at ∞ is undefined ... 1/ ∞

... but we do know that 1/x approaches0 as x approaches infinity

Limits Approaching InfinityWhat is the limit of this function?

y = 2x

Obviously as "x" gets larger, so does "2x":

x y=2x

1 2



2 4

4 8

10 20

100 200

... ...

So as "x" approaches infinity, then "2x" also approaches infinity. We writethis:

But don't be fooled by the "=". You cannot actually get toinfinity, but in "limit" language the limit is infinity (which isreally saying the function is limitless).

Infinity and DegreeWe have seen two examples, one went to 0, the other went to infinity.

In fact many infinite limits are actually quite easy to work out, if you canfigure out "which way it is going", like this

Functions like 1/x approach 0 as x approaches infinity. This isalso true for 1/x 2etc

A function such as x will approach infinity, as well as 2x , or x/9and so on. Likewise functions with x 2 or x 3 etc will also approachinfinity

But be careful, a function like " -x " will approach " -infinity ", soyou have to look at the signs of x .

In fact, if we look at the Degree of the function (the highest





exponent in the function) we can tell what is going tohappen:

If the Degree of the function is:• greater than 0, the limit is infinity (or -infinity)

• less than 0, the limit is 0

But if the Degree is 0 or unknown then we need to work a bit harderto find a limit

Rational Functions

A Rational Function is one that is the ratio of two polynomials:

For example, here P(x)=x 3 +2x-1 , and Q(x)=6x 2 :

Following on from our idea of the Degree of the Equation , the first step to

find the limit is to ...Compare the Degree of P(x) to the Degree of Q(x) :

If the Degree of P is less than the Degree of Q ...

... the limit is 0.

If the Degree of P and Q are the same ...... divide the coefficients of the terms with the largest exponent , like this:









If the Degree of P is greater than the Degree of Q ...

... then the limit is positive infinity ...

... or maybe negative infinity. You need to look at the signs!

You can work out the sign (positive or negative) by looking at the signs of the terms with the largest exponent , just like how we found the coefficientsabove:

For example this will go to positive infinity, becauseboth ...

• x 3(the term with the largest exponent in the top) and

• 6x 2(the term with the largest exponent in the bottom)

... are positive.

But this will head for negative infinity, because -2/5 is negative.

A Harder Example: Working Out "e"There is a formula for the value of e (Euler's number) based on infinityand this formula:

(1+ 1/n) n

http://www.mathsisfun.com/numbers/e-eulers-number.html




At Infinity: (1+1/ ∞ ) ∞ = ??? ... we don't know!

So instead of trying to work it out for infinity (because we can't get asensible answer), let's try larger and larger values of n:

n (1 + 1/n)n

1 2.00000

2 2.25000

5 2.48832

10 2.59374

100 2.70481

1,000 2.71692

10,000 2.71815

100,000 2.71827

It settles down to a value (2.71828... which is the magic number e )

So again we have an odd situation:• We don't know what the value is when n=infinity

• But we can see that it settles towards 2.71828...

So, we use limits to write the answer like this:

It is a mathematical way of saying "we are not talking about when n= ∞, but we know as n gets bigger, the answer gets closer and closer to the value of e " .

Don't Do It The Wrong Way ... !

You can see by the graph and the table that as n get largerthe function approaches 2.71828....But trying to use infinity as a "very large real number" ( it isn't! ) would give this:

(1+1/ ∞) ∞ = (1+0) ∞ = (1) ∞ = 1

So, don't try to use Infinity as a real number, you will get





wrong answers !

Limits are the right way to go.

Evaluating LimitsI have taken a gentle approach to limits so far, and shown tables and graphsto illustrate the points.

But to "evaluate" (in other words calculate) the value of a limit can take a bitmore effort. I will show you how in Evaluating Limits .

What is Infinity?

Infinity ...

http://www.mathsisfun.com/calculus/limits-evaluating.html

http://www.mathsisfun.com/calculus/limits-evaluating.html



... it's not big ...

... it's not huge ...

... it's not tremendously large ...

... it's not extremely humongously enormous ...

... it's ...

Endless!

Infinity has no endInfinity is the idea of something that has no end.

In our world we don't have anything like it. So we imagine traveling on and on, trying hard to getthere, but that is not actually infinity.

So don't think like that (it just hurts your brain!). Just think "endless", or "boundless".

If there is no reason something should stop, then it is infinite.

Examples:

{1, 2, 3, ...} The sequence of natural numbers never ends, and is infinite.

OK, 1/3 is a finite number (it is not infinite). But written as a decimalnumber the digit 3 repeats forever (we say "0.3 repeating"):

0.3333333... (etc)

There's no reason why the 3s should ever stop: they repeat infinitely .

http://www.mathsisfun.com/whole-numbers.html





0.999...So, when you see a number like "0.999..." (i.e. a decimal number withan infinite series of 9s), there is no end to the number of 9s.

You cannot say "but what happens if it ends in an 8?", because it simplydoes not end. (This is why 0.999... equals 1).

AAAA... An infinite series of "A"s followed by a "B" would NEVER have a "B".

There are infinite points in a line. Even a short line segmenthas infinite points.

Infinity does not growInfinity is not "getting larger", it is already fully formed.

Sometimes people (including me) say it "goes on and on" which sounds like it is growingsomehow. But infinity does not do anything, it just is.

Infinity is not a real numberInfinity is not a real number, it is an idea. Anidea of something without an end.

Infinity cannot be measured.

Even these faraway galaxies can't compete withinfinity.

Infinity is SimpleYes! It is actually simpler than things which do have an end. Because if something has an end,you have to define where that end is.

Example: in Geometry a "Line" has infinite length ... it goes in bothdirections without an end.

If it has one end it is called a Ray, and if it has two ends it is called aLine Segment, but that needs extra information to define where theends are.

Big NumbersThere are some really impressively big numbers.



A Googol is 1 followed by one hundred zeros (10 100) :

10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

A Googol is already bigger than the number of elementary particles in the known Universe, butthen there is the Googolplex . It is 1 followed by Googol zeros . I can't even write down thenumber, because there is not enough matter in the universe to form all the zeros:

10,000,000,000,000,000,000,000,000,000,000,000,000,...etc (Googol number of Zeros)

And there are even larger numbers that need to use "Power Towers" to write them down.

For example, a Googolplex can be written as this power tower:That is ten to the power of (10 to the power of 100),

But imagine an even bigger number like

And you can easily create much larger numbers than those!

FiniteAll of these numbers are "finite", you could eventually "get there".

But none of these numbers are even close to infinity. Because they are finite, and infinity is ...notfinite!

Using InfinityWe can sometimes use infinity like it is a number, but infinity does not behave like a realnumber.

To help you understand, think "endless" whenever you see the infinity symbol " ∞":For example: ∞ + 1 = ∞

Which says that infinity plus one is still equal to infinity.

If something is already endless, you can add 1 and it will still be endless.

The most important thing about infinity is that:

-∞ < x < ∞

Where x is a real

number

Which is mathematical shorthand for " minus infinity is less than any real number,and infinity is greater than any real number"

Here are some more properties

Special Properties of Infinity

http://www.mathsisfun.com/numbers/real-numbers.html






∞ + ∞ = ∞

-∞ + - ∞ = - ∞

∞ × ∞ = ∞

-∞ × - ∞ = ∞

-∞ × ∞ = - ∞

x + ∞ = ∞

x + (- ∞ ) = - ∞

x - ∞ = - ∞

x - (- ∞ ) = ∞

For x >0 :

x × ∞ = ∞

x × (- ∞ ) = -∞

For x <0 :

x × ∞ = -∞

x × (- ∞ ) = ∞

Undefined OperationsAll of these are "undefined":

"Undefined" Operations

0 × ∞



0 × - ∞

∞ + - ∞

∞ - ∞

∞ / ∞

∞ 0

1 ∞

Example: Isn't ∞ / ∞ equal to 1?

No, because we really don't know how big infinity is, so we can't say that two infinities are thesame. For example ∞ + ∞ = ∞ , so

∞

=

∞ +

∞ which would mean that:

1

=

2

∞ ∞ 1 1

And that doesn't make sense! I could have also made 1=3 and so on ... so we say that ∞ / ∞ is

undefined.Infinite SetsIf you continue to study this subject you will find discussions about infinite sets, and the idea of different sizes of infinity.

That subject has special names like Aleph-null (how many Natural Numbers), Aleph-one and soon, which are used to measure the sizes of sets .

For example, there are infinitely many whole numbers {0,1,2,3,4,...}, but there are more realnumbers (such as 12.308 or 1.1111115) because there are infinitely many possible variationsafter the decimal place as well.

But that is an advanced topic, and goes beyond the simple concept of infinity we discuss here.ConclusionInfinity is a simple idea: "endless". Most things we know have an end, but infinity does not.

http://www.mathsisfun.com/sets/sets-introduction.html











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