022 Sample Final A, Problem 14

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Find the following limit: .

When evaluating limits of rational functions, the first idea to try is to simply plug in the limit. In addition to this, we must consider that as a limit,
In the latter case, the sign matters. Unfortunately, most (but not all) exam questions require more work. Many of them will evaluate to an indeterminate form, or something of the form


In this case, there are several approaches to try:
  • We can multiply the numerator and denominator by the conjugate of the denominator. This frequently results in a term that cancels, allowing us to then just plug in our limit value.
  • We can factor a term creatively. For example, can be factored as  , or as  , both of which could result in a factor that cancels in our fraction.
  • We can apply l'Hôpital's Rule: Suppose is contained in some interval . If   and   exists, and   for all   in , then .
Note that the first requirement in l'Hôpital's Rule is that the fraction must be an indeterminate form. This should be shown in your answer for any exam question.


Step 1: 
We take the limit and find that
This is an indeterminate form, and we need to apply l'Hôpital's Rule.
Step 2: 
Applying l'Hôpital's Rule (by taking the derivative of the numerator and denominator separately), we find:
Final Answer:  

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