Calculus 2 : Finding Limits and One-Sided Limits

Study concepts, example questions & explanations for Calculus 2

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Example Questions

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Example Question #1 : Finding Limits And One Sided Limits

Which graph is a possible sketch of the function  that possesses the following characteristics: 

Possible Answers:

Graph2

There does not exist such a graph.


Graph4

Graph1

Graph3

Correct answer:

Graph1

Explanation:

Since  there is a possible vertical asymptote at .

As we approach from the left, the graph should tend to . Approaching  from the right, the graph tends to .

The only graph that does so is  Graph1.

Example Question #45 : Calculus Ii

Rational_graph

The graph above is a sketch of the function . Find .

Possible Answers:

Does not exist

Correct answer:

Explanation:

We need to look at the behavior of the function as it tends to  from the left.  Therefore the answer is

Example Question #2 : Finding Limits And One Sided Limits

Rational_graph

The graph above is a sketch of . Find .

Possible Answers:

Does not exist

Correct answer:

Does not exist

Explanation:

The limit does not exist because the one-sided limits are not equal; , whereas 

Example Question #3 : Finding Limits And One Sided Limits

Determine the value of .

Possible Answers:

Correct answer:

Explanation:

The expression  indicates that all points on the domain are equal to 5 since the absolute value negates negative values.

The  is to search for the limit as the graph approaches  from the left side of the graph.  Since the absolute value of negative five is five, the graph approaches five from the left.

The correct answer is .

Example Question #4 : Finding Limits And One Sided Limits

Evaluate the following limit.

Possible Answers:

Limit does not exist

Correct answer:

Explanation:

This limit can be solved using simple manipulation of the expression inside the limit:

Example Question #5 : Finding Limits And One Sided Limits

Evaluatle the limit:

Possible Answers:

Correct answer:

Explanation:

Consider the domain of the function. Because this equation is a polynomial, x is not restricted by any value. Thus the way to evaluate this limit would simply be to plug the value that x is approaching into the limit equation.

Example Question #6 : Finding Limits And One Sided Limits

Evaluate the limit:

Possible Answers:

Correct answer:

Explanation:

Consider the domain of the function. Because this equation is a polynomial, x is not restricted by any value. Thus the way to evaluate this limit would simply be to plug the value that x is approaching into the limit equation.

Example Question #51 : Calculus Ii

Evaluate the limit:

Possible Answers:

Correct answer:

Explanation:

The limiting situation in this equation would be the denominator. Plug the value that x is approaching into the denominator to see if the denominator will equal 0. In this question, the denominator will not equal zero when x=-2; so we proceed to insert the value of x into the entire equation.

Example Question #52 : Calculus Ii

Evaluate the limit:

Possible Answers:

Correct answer:

Explanation:

The limiting situation in this equation would be the denominator. Plug the value that x is approaching into the denominator to see if the denominator will equal 0. In this question, the denominator will not equal zero when x=0; so we proceed to insert the value of x into the entire equation.

Example Question #53 : Calculus Ii

Evaluate the limit:

Possible Answers:

Correct answer:

Explanation:

The limiting situation in this equation would be the denominator. Plug the value that x is approaching into the denominator to see if the denominator will equal 0. In this question, the denominator will equal zero when x=5; so we try to eliminate the denominator by factoring. When the denominator is no longer zero, we may continue to insert the value of x into the remaining equation.

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