Calculus 2 : Limits

Study concepts, example questions & explanations for Calculus 2

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Example Questions

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Example Question #1 : Calculus Ii

Evaluate:

Possible Answers:

None of the other answers is correct.

The limit does not exist.

Correct answer:

None of the other answers is correct.

Explanation:

Evaluated at , the numerator and the denominator are both equal to 0, as shown below:

So a straightforward substitution will not work. L'Hospital's rule will work here, but an easier way is to note that

 and .

So the expression can be rewritten - and solved - as follows:

Example Question #1 : Limit Concepts

Evaluate the limit:

Possible Answers:

Does Not Exist

Correct answer:

Explanation:

Directly evaluating the limit will produce an indeterminant answer of .

Rewriting the limit in terms of sine and cosine, , we can try to manipulate the function in order to utilize the property .

Multiplying the function by the arguments of the sine functions, , we can see that the limit will be .

 

 

 

Example Question #1 : Limits

Evaluate .

Possible Answers:

The limit does not exist.

Correct answer:

Explanation:

and 

,

so we cannot solve this by substituting.

However, we can rewrite the expression:

Example Question #4 : Calculus Ii

Find the limit of  as  approaches infinity.

Possible Answers:

Inconclusive

Correct answer:

Explanation:

The expression  can be rewritten as .

Recall the Squeeze theorem can be used to solve for the limit.  The sine function has a range from , which means that the range must be inside this boundary.

Multiply the  term through.

Take the limit as  approaches infinity for all terms.

Since the left and right ends of this interval are zero, it can be concluded that  must also approach to zero.

The correct answer is 0.

Example Question #1 : Limit Concepts

Determine the limit.   

Possible Answers:

Correct answer:

Explanation:

To determine, , graph the function  and notice the direction from the left and right of the curve as it approaches .

Both the left and right direction goes to negative infinity.

The answer is:  

Example Question #6 : Calculus Ii

Which of the following is true?

Possible Answers:

If neither  nor  exist, then  also doesn't exist.

If  exists, then  and  both exist.

 and  exist if and only if  exists.

If  and , then  exists.

Correct answer:

If  and , then  exists.

Explanation:

If  and , then  exists.

This can be proven rigorously using the  definition of a limit, but it is most likely beyond the scope of your class. 

Example Question #7 : Calculus Ii

Determine the limit:  

Possible Answers:

Correct answer:

Explanation:

Isolate the constant in the limit.

The limit property .

Therefore:

Example Question #8 : Calculus Ii

Evaluate the limit, if possible:  

Possible Answers:

Correct answer:

Explanation:

To evaluate , notice that the inside term  will approach infinity after substitution.  The inverse tangent of a very large number approaches to .

The answer is .

Example Question #9 : Calculus Ii

Evaluate the following limit:

Possible Answers:

Correct answer:

Explanation:

The first step is to factor out the highest degree term from the polynomial on top and bottom (essentially pulling out 1):

which becomes

Evaluating the limit, we approach .

 

Example Question #10 : Calculus Ii

Evaluate the following limit:

Possible Answers:

Correct answer:

Explanation:

To evaluate the limit, first pull out the largest power term from top and bottom (so we are removing 1, in essence):

which becomes

Plugging in infinity, we find that the numerator approaches zero, which makes the entire limit approach 0.

 

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