MAT01A1: Limits

Dr Craig

28 February 2018

Note: these lecture slides should be inused in conjunction with the textbook.These slides do not tell the full story.You must read the sections from thetextbook and complete the prescribedtutorial exercises in order to fullyunderstand each section.

Semester Test 1

I Saturday 10 March at 08h30

I See detailed announcement on

Blackboard

Saturday class this week

I 09h00 – 12h00

I C-LES 402

e-Quiz 3

I Opens Thursday afternoon.

I Closes Friday 9 March at 23h59.

I Covers Chapter 1.

Tut 5

Questions are available under “Tutorials”.

Past paper

Semester Test 1 from 2017 available under

“Assessments”.

Some inverse trig examples

Calculate the following

I arcsin(−√3

2

)I cos(arcsin t)

I cos(arctan 23)

I Tough one:

sec(arcsin(2x− 1))

Today’s lecture: Limits

I Why limits?

I The definition of a limit

I Estimating limits by calculation

I One-sided limits

I Infinite limits

I Vertical asymptotes

I Khan Academy exercises on limits

The tangent to a function

Consider the parabola

y = x2

We are interested in the tangent to f (x) at

the point P = (1, 1).

We take different values of Q (with Q 6= 1)

and calculate the gradient of the straight line

that goes through P and Q.

Values of mPQ

x < 1 mPQ

0 1

0.5 1.5

0.9 1.9

0.99 1.99

0.999 1.999

x > 1 f (x)

2 3

1.5 2.5

1.1 2.1

1.01 2.01

1.001 2.001

The tangent line as Q approaches P

Playing around with limits

Go to the Stewart Calculus link:

Tools for enriching calculus

Click on “Limits and Derivatives”, then click

on “Secant Line and Tangent Line”. The red

dot (x = a) represents the fixed point P and

by adjusting the value of h you can move the

blue dot Q.

Velocity

We can easily measure distance with respect

to time, but to measure instantaneous

velocity we need to use limits.

Dropping a ball from a tower

Definition of a limit of a function

Definition: Suppose f(x) is defined for x near*a. If we can get the values of f(x) as close to Las we like by taking x as close to a on either side(but not equal to a), then we write

limx→a

f(x) = Land say“the limit of f(x), as x approaches a, equals L”

*near: f(x) is defined on some open intervalincluding a, but possibly not at a itself.

Example

Estimate the value of limx→1

x− 1

x2 − 1.

x < 1 f(x)

0.5 0.666667

0.9 0.526316

0.99 0.502513

0.999 0.500250

0.9999 0.500025

x > 1 f(x)

1.5 0.4

1.1 0.476190

1.01 0.497512

1.001 0.499750

1.0001 0.499975

Based on the above calculations, we estimate that

limx→1

x− 1

x2 − 1=

1

2.

Example:

Consider the function f (x) =x2 + 4x + 4

x + 2.

Observe that

f (x) =(x + 2)(x + 2)

x + 2

Is this different from the function

g(x) = x + 2?

Example

g(x) =

{x + 3 if x 6= 2

1 if x = 2

Sketch the function and find limx→2

g(x).

One-sided limits

Definition: We write

limx→a−

f(x) = L

and say “the limit of f(x) as x approaches afrom the left” is equal to L if we can make thevalues of f(x) as close as we like to L by takingx close to a, but with x less than a.

This is also called “the left-hand limit of f(x) as xapproaches a”

From the other side

Definition: We write

limx→a+

f(x) = L

and say “the limit of f(x) as x approaches afrom the right” is equal to L if we can make thevalues of f(x) as close as we like to L by takingx close to a, but with x greater than a.

This is also called “the right-hand limit of f(x) as xapproaches a”

One-sided example

h(x) =

cosx if x < 012 if x = 0

x if x > 0

limx→0−

h(x) = 1, limx→0+

h(x) = 0, h(0) =1

2

One (sided) + one (sided) = two (sided)

For L ∈ R we have the following:

limx→a

f (x) = L if and only if

limx→a−

f (x) = L and limx→a+

f (x) = L

If there exists L ∈ R such that

limx→a

f (x) = L then we say:

“the limit of f (x) as x approaches a exists”

One (sided) + one (sided) = two (sided)

In the previous two examples of piece-wise

functions we have:

I limx→2

g(x) exists

(since limx→2−

g(x) = 5 = limx→2+

g(x))

I limx→0

h(x) does not exist

(because limx→0−

h(x) 6= limx→0+

h(x))

Infinite limits:

Definition: Let f be a function defined on

both sides of a, except possibly at a itself.

Then

limx→a

f (x) =∞

means that f (x) can get as big as we like

by taking x sufficiently close to a.

Example: limx→0

1

x2=∞

One-sided infinite limits:

Let f (x) =1

x,

and consider

limx→0−

f (x) and limx→0+

f (x)

Let g(x) = cscx, and consider

limx→π−

g(x) and limx→π+

g(x)

Very important:

For the limit at x approaches a of f (x) to

exist, we need

limx→a

f (x) = L

for some L ∈ R. Therefore, limx→0

1

x2does not

exist.

Example: find the following limits:

limx→3+

2x

x− 3and lim

x→3−

2x

x− 3

To calculate the first limit, consider an x

value just a little bit bigger than 3 (e.g. 3.1)

and then see what happens as you get closer

to 3 (e.g. 3.01).

To calculate the second limit, consider an x

value slightly smaller than 3 (e.g. 2.9) and

then get closer to 3 (e.g. 2.99).

Vertical asymptotes

The line x = a is called a verticalasymptote of the curve y = f (x) if at

least one of the following statements is

true:

limx→a

f (x) =∞ limx→a

f (x) = −∞

limx→a−

f (x) =∞ limx→a−

f (x) = −∞

limx→a+

f (x) =∞ limx→a+

f (x) = −∞

Important: as stated on the previous slide, youonly need one of the listed conditions to be satisfiedin order to have a vertical asymptote at x = a; e.g.:

f(x) =

{1 if x 6 0

lnx if x > 0

limx→0+

f(x) = −∞ so f(x) has a V.A. at x = 0.

Example

Find the vertical asymptotes of the following

function:

f (x) =x2 + 2

4x + 4x2

You should find two x values where f (x)

might have a vertical asymptote. Call these

values a and b. Now find

limx→a+

f (x) and limx→a−

f (x)

and

limx→b+

f (x) and limx→b−

f (x)

Khan Academy exercises on limits

Click on the hyperlinks to test your

understanding of limits:

One-sided limits from graphs

Two-sided limits from graphs