Question17.
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SCALCET9M 4.4.053.
Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it.
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STEP 1
Assumptions
1. We need to find the limit as approaches .
2. The expression given is .
3. We will use l'Hospital's Rule if the limit results in an indeterminate form like or .
4. If a simpler method is available, we will consider using it.
STEP 2
First, let's analyze the expression to see if it results in an indeterminate form as .
As :
-
- (using the first-order Taylor expansion of around 0)
- Therefore,
Thus, the expression becomes , which is an indeterminate form.
STEP 3
To resolve the indeterminate form, we will combine the terms into a single fraction:
STEP 4
Now, evaluate the limit of the new expression:
As , both the numerator and the denominator approach 0, resulting in a indeterminate form. This allows us to apply l'Hospital's Rule.
STEP 5
Apply l'Hospital's Rule by differentiating the numerator and the denominator:
Numerator:
Denominator:
STEP 6
Apply l'Hospital's Rule:
STEP 7
Evaluate the limit:
As , the expression simplifies to:
Thus, the limit is:
The limit is .
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