L'Hospital's Rule

Please see the attached file for the fully formatted problems.

17 L`Hospital`s Rule
These are instructions from my professor he wants us to use this form & these rules to do the problems. He wants us to follow these sepd. Please help, I appreciate it. There are 3 problems in this set and 3 pages. thx
Note this is not the Quotient Rule. The rule states that if certain conditions are satisfied, the limit of a fraction can be computed by calculating the limit of the fraction of the derivatives. It is important to always check if one can apply the the rule. L'Hospital's Rule
The rule is useful when calculating some limits of fractions for which the Limit Law: cannot be applied because for example and as in this limit: .
Theorem (L'Hospital's Rule). Suppose
and or that
and . Then

if the limit on the right side exists.

Wrong: Why? Example
In this limit , both the denominator and the numerator tend to zero as . Hence, we can apply L'Hospital's Rule.

Other limits
In order to apply L'Hospital's Rule to other limits, we need to convert them to fractions. The easiest is to convert a product. or if does not work try . For example,

The sum and difference can be converted to products or . Finally, powers can be converted to products by using the continuity of the exponential function. .

Homework
Before submitting your homework, check all problems by graphing the function near the limit and see if the values of support your solution. In each problem for which you would like to use L' Hospital's Rule, when you convert your expression to a fraction, check and state that the hypotheses of the theorem are satifsied.
Find limits. Use L'Hospital's Rule when appropriate. If there is a more elementary method, you might consider it.
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