Calculus 2 : Finding Limits and One-Sided Limits

Study concepts, example questions & explanations for Calculus 2

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Example Questions

Example Question #331 : Finding Limits And One Sided Limits

Evaluate the limit:

Possible Answers:

Correct answer:

Explanation:

To evaluate the limit, we can use properties of exponents and factorials to simplify:

The denominator goes to infinity as n goes to infinity, which makes the whole term go to zero.

Example Question #332 : Finding Limits And One Sided Limits

Evaluate the limit:

Possible Answers:

Correct answer:

Explanation:

When evaluating the limit, we must consider the terms in the limit. While the numerator and denominator both have a polynomial term of the same degree (which would suggest dividing their coefficients to get the limit), there is also an exponential term in the denominator, which grows faster than the polynomial terms. The denominator reaches infinity faster, which means the limit equals  (the overall term has a far larger term in the denominator than the numerator).

Example Question #333 : Finding Limits And One Sided Limits

Evaluate the limit:

Possible Answers:

Correct answer:

Explanation:

To evaluate the limit, we must factor out a term consisting of the highest power term divided by itself (which equals one, so we aren't changing the original function):

The term we factored goes to one, and the two terms with negative exponents in the denominator go to zero (they are each "fractions" with n in their denominator - the terms go to zero as the denominator goes to infinity), so we are left with .

Example Question #334 : Finding Limits And One Sided Limits

Evaluate the limit:

Possible Answers:

Correct answer:

Explanation:

To evaluate the limit, we can simplify the function using properties of factorials and exponents:

The simplified function has n only in the numerator; the function goes to infinity as n goes to infinity.

Example Question #335 : Finding Limits And One Sided Limits

Evaluate the limit:

Possible Answers:

Correct answer:

Explanation:

To evaluate the limit, we must first check to see from which side we are evaluating the limit. The negative sign exponent indicates that we are evaluating the limit from the left side, or using numbers smaller than 2. This corresponds to the first part of the piecewise function. When we evaluate the limit using this function, we get .

Example Question #336 : Finding Limits And One Sided Limits

Evaluate the limit:

Possible Answers:

Correct answer:

Explanation:

To evaluate the limit, we must compare the terms in the numerator and denominator. The numerator and denominator both have polynomial terms of the same degree. However, the numerator also has an exponential term that grows far faster than the polynomial terms. So, its growth dominates and the numerator goes to infinity, thus the limit of the overall function is .

Example Question #337 : Finding Limits And One Sided Limits

Evaluate:

Possible Answers:

Correct answer:

Explanation:

To evaluate the limit, we must use the properties of exponents and factorials to simplify the function:

As n goes to infinity, the denominator of the simplified function gets infinitely large, which means the term goes to zero.

 

Example Question #381 : Limits

Evaluate:

Possible Answers:

Correct answer:

Explanation:

To evaluate the limit, we must factor out a term consisting of the highest power term divided by itself (which is the same as 1, so we aren't changing the original function):

After the term we factored out cancels to one, and the negative power terms in the denominator go to zero (as n goes to infinity, these terms contain infinity in their denominator, which makes the whole term go to zero), we are left with

Example Question #382 : Limits

If it exists, find the following limit

Possible Answers:

Does not exist

Correct answer:

Explanation:

To find the following limit

we can simply plug in  because it's defined at that point and in infinitesimally small neighborhoods near , so we get

Example Question #383 : Limits

Find the following limit

Possible Answers:

Correct answer:

Explanation:

To find the following limit

we can see that  approaches  as , so we get the fraction , which approaches , so we get

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