Calculus 2 : Limits and Asymptotes

Study concepts, example questions & explanations for Calculus 2

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Example Questions

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Example Question #1 : Limits And Asymptotes

Find the vertical asymptotes of the function 

Possible Answers:

There are no vertical asymptotes.

Correct answer:

Explanation:

A vertical asymptote occurs at  when 

or .

In our case, since we have a quotient of functions, we need only check for values of  that make the denominator , but don't also make the numerator 



This equals  when  is an integer multiple of .

Hence the vertical lines  are vertical asymptotes.

However we must exclude the case , because this will also cause the numerator to be , thus creating a "hole" instead of an asymptote.

Hence our answer is

 

.

Example Question #2 : Limits And Asymptotes

What is the value of the limit of the function below: 

Possible Answers:

Correct answer:

Explanation:

We note that for all , we have .

Hence,

 

By inverting the above inequality and multiplying by x. We get the following:

 

.

 

We know that,

and by the Squeeze Theorem,

we have:

Example Question #3 : Limits And Asymptotes

How many vertical asymptotes does the following function have?

Possible Answers:

It does not have a vertical asymptote.

It has only one vertical asymptote.

 

The function has infinitely many vertical asymptotes.

Correct answer:

The function has infinitely many vertical asymptotes.

Explanation:

We first need to see that the function sin(x) has infinitely many roots.

We can express these roots in the following form:

, wkere k is an integer.

The function has the roots as asymptotes.

Therefore this function's vertical asymptotes are expresses by , where k is an integer. Since the integers are infinitely many, the vertical asymptotes are infinitely many.

Example Question #4 : Limits And Asymptotes

Find the following limit:

  , where  is positive integer.

Possible Answers:

The limit does not exist.

Correct answer:

Explanation:

To find the above limit, we need to note the following.

We have for all n positive integers:

.

(We can verify this formula by the long division)

Now we need to note that:

, where .

We have then:

 

and we have

.

Since,

 

 

we obtain the following:

 

Example Question #5 : Limits And Asymptotes

How many asymptotes does the function below have: 

 is assumed to be a positive ineteger.

 

 

Possible Answers:

It has infinitely many

Correct answer:

Explanation:

We need to notice that the function f is defined for all real numbers.

We need to also remark that for all reals:

implies that

this gives again:

and therefore,

.

This function can't be 0.

Assume for a moment that

, this implies that but this cannot happen since we are dealing with real numbers.

Therefore the above function can never be 0 and this means that it does not have a vertical asymptote. This is what we needed to show.

 

 

 

 

Example Question #6 : Limits And Asymptotes

Find the following limit:

Possible Answers:

Correct answer:

Explanation:

For this problem we first need to expand the denominator.

We can expand the denominator since  is a difference of squares.

From here we can cancel the  quantity from the numerator and denominator.

The resulting function is as follows:

Plugging in 2 we get our limit.

 

Example Question #7 : Limits And Asymptotes

Find the following limit:

 

Possible Answers:

Correct answer:

Explanation:

We will use the following to prove this result.

Assuming that

. We will use this result:

 

we have

Therefore

this shows the limit is 1.

Example Question #8 : Limits And Asymptotes

Find the following limit:

Possible Answers:

The limit does not exist.

Correct answer:

Explanation:

We will use the following identity to establish this result.

We have

and we note that :

Therefore by multiplying the above equivalency for 1 we get the following:

 and we know that

 

We can rewrite our equation using identities. 

This gives :

 and

now taking the limit as x goes to 3, we obtian

 

 

Example Question #9 : Limits And Asymptotes

Find the following limit:

Possible Answers:

Correct answer:

Explanation:

We note first that we can write:

Therefore our expression becomes in this case:

Noting now that:

for all .

 

Therefore , we have: 

and evaluating now for x=1, we obtain

 

Example Question #10 : Limits And Asymptotes

Let  the following polynomial:

What are the vertical asymptotes of

 

Possible Answers:

 does not have a vertical asymptote.

There are  vertical asymptotes

Correct answer:

 does not have a vertical asymptote.

Explanation:

We first note that the polynomial is defined for all real numbers.

We know that for any real number x different from 0, we have :

.

 

Now we need to see that for any integer n we have:

. Adding in this case,

we have

and therefore , this implies by definiton of q(x) that:

.

We also have .

This means that  .

Therefore q(x) can never be 0 and this means that it does not have an asymptote.

 

 

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