What is a limit?

COVID Year 8 limits, continuity, IVT.notebook

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What is a limit?


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So according to the College Board students will know that...

Given a function f, the limit of f(x) as x approaches c is a real number R if f(x) can be made arbitrarily close to R by taking x sufficiently close to c (but not equal to c). If the limit exists and is a real number, then the common notation is

Hmmm...I am puzzled by this! What about piecewise functions???


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Infinite LimitsConsider the behavior of f(x) = 1/x for values of x near 0.

What do the following expressions mean?

and

and

Graphically what do these limits imply?


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Limits at InfinityConsider the behavior of f(x) = 1/x when:

a) x increases without bound (x )

b) x decreases without bound (x )

Graphically what do these limits imply?


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Famous Limit at InfinityLet


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What about basic trig functions?Consider the graph of f(x) = sin x

Evaluate:


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When would limits not exist?

1)

2)

3)

https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=5&cad=rja&uact=8&ved=0ahUKEwi776PUsP3OAhVD5SYKHbmQBCYQtwIIKzAE&url=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3DoDAKKQuBtDo&usg=AFQjCNFRRblXa6oCPMmySwddC5Sf1Mhz2Q

https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=5&cad=rja&uact=8&ved=0ahUKEwi776PUsP3OAhVD5SYKHbmQBCYQtwIIKzAE&url=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3DoDAKKQuBtDo&usg=AFQjCNFRRblXa6oCPMmySwddC5Sf1Mhz2Q


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Evaluate each by any means necessary!!!!

Find

Find


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All reals, x cannot equal 1

Line y = x 4, with a hole at x = 1

5

5


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5

The intended destination from left and right


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Basic LimitsLet a and k represent real numbers.

1)

2)

3)

4)


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Evaluate:


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Classwork Page 67 #55


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Evaluate the following limit in two different ways,


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Homework

page 67 #56

page 76 #2, 13


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Limits of Polynomial Functions as x approaches ±∞Consider the graphs of each and exam the limits as x ±∞ a) y = x

b) y = x2

c) y = x3

d) y = x4

e) y = x2

What general results follow from these special cases?


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Ex: Evaluate the following limits


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Limits of Piecewise FunctionsWe determine the limits of piecewise functions by considering the one sided limits.

Example:

Evaluate: (a)

(b)

(c)

First graph then evaluate.


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Rational FunctionsFind

Find

Find


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Find

Theorem:Let be a rational function, and let a be any real number.

(a) If q(a) ≠ 0, then

(b) If q(a) = 0 but p(a) ≠ 0, then

(c) If q(a) = 0 = p(a), then


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Rational Functions in which the limit of the denominator = 0and as x ±∞


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Find

What do you think the limit will be?


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Find What do you think?


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Find What do you think?


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What are these limits really finding?


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Limits Involving RadicalsEvaluate What do you think?


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Find


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7) Find


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8) Find


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9) Find


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Before we begin the next example let's review a fun fact, shall we?

=

Find


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10) Find


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Evaluate:


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4/3 0 2

8 12 0

4/5 4/3


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2

DNE 1/4

k = 0

neg. infinity infinity

1.5 0

infinity

1


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Limits of Exponential Functions


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Rookie Mistakes Practice1)

2)

3)


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End Behavior of Logarithmic Functions


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Absolute ValueGiven, find


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2

2

1/4

2

1

2

1


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1/2

1/2

DNE

infinity


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Trigonometric Functions


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Memorization Card


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Example:

Evaluate

What do you think?


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Trig Limits


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Memorization Card


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What do you think?


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What do you think?

Update memorization card.


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Evaluate.


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What do you think?


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What do you think?


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What do you think?


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What do you think? What do you think?


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Evaluate the two limits

Evaluate the limit

Evaluate the limit

Evaluate the limit

Evaluate the limit


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algebraically


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Hierarchy of DominanceGraph the following functions:

i(x) = x2f(x) = xxj(x) = ln(x)h(x) = exg(x) = x!

If all else fails....


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Example:

Given , is this function continuous at x = 1?


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Example:

Is this function continuous at x = 3?


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Example:Find the value of k that would make the following function continuous at x = 5.


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Example:Find the value of k that would make the following function continuous at x = 3.


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Example:Is the function continuous at x = 0?


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Example:For what values of x is there a discontinuity in the graph of:


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Example:Show that f(x) = is continuous everywhere.


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Continuity on an Interval

If a function, f(x), is continuous at each number in an open interval (a,b), then we say f(x) is __________________________________.

This also applies to infinite open intervals like (∞, a), etc.

If f(x) is continous on (∞, ∞) we say that f(x) is _________________

_____________________.

Example:


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What about on a closed interval?

For a function to be continuous at an endpoint of an interval its value at the endpoint must be equal to the appropriate onesided limit at that endpoint.

In other words:


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Example:What can you say about the continuity of the function ?


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Continuity of CompositionsWhy even learn this?It will be useful for calculating limits of compositions of functions.

Theorem:If and if the function f is continuous at L, then

Theorem:

1. If the function g is continuous at c, and the function f is continuous at g(c), then the composition f(g(x)) is continuous at c.

2. If the function g is continuous everywhere and the function f is continuous everywhere, then the composition f(g(x)) is continuous everywhere.


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Example: What do you think?


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Pushing the IVT further...

Given [a, b], I would like you to plot the points

(a, f(a)) and (b, f(b)) such that:

1) Pick a height in which f(a)>0 and f(b) <0.

2) Draw a continuous function from f(a) to f(b).

3) We will compare it to mine.


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My graph is below.

Do we have the exact same graph?

What do you notice about the two? Is there anything they have in common?


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Memorization Card


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continuous


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Homework

Page 66

30)

33)

Page 76

1)

3)

4)

Page 97

2)

5)

Page 8586

14) At what values of x is f continuous.

47) Find a value for a so that the function

is continuous.

58) Which of the following statements about the function

is not true?

(A) f(1) does not exist

(B) exists

(C) exists

(D) exists

(E)

=2

=(1)(0)=0

=5/21

[1,0) U (0, 1) U (1, 2) U (2, 3)

x = 4/3

Choice A

1


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Homework

page 77 #69

page 9798 #10, 13, 40

10) Find the limit:

Find the limit

13) Find the limit:

40) What value should be assigned to k to make f a continuous function?

=2

= 0

= 0

k = 1/2


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What is a limit?