Math Problem Statement
Solution
To solve this limit:
we need to examine the behavior of each component as .
Step-by-Step Solution
-
Analyze the components of the expression:
- As , , because is close to for small values of .
- Therefore, as .
-
Rewrite the expression:
- The limit can now be expressed as:
-
Identify the indeterminate form:
- As , and . Thus, is of the indeterminate form .
-
Rewrite using L'Hôpital's Rule:
- To apply L'Hôpital's Rule, rewrite as .
- Then, the limit becomes:
- This is now in the form , which is suitable for L'Hôpital's Rule.
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Apply L'Hôpital's Rule:
- Differentiate the numerator and the denominator with respect to :
- The derivative of is .
- The derivative of is .
- Thus, we have:
- Differentiate the numerator and the denominator with respect to :
-
Evaluate the final limit:
- As , .
- Therefore,
Final Answer
Would you like more details on any step, or do you have additional questions?
Here are five related questions to expand your understanding:
- What is the purpose of using L'Hôpital's Rule in solving limits?
- Why is a valid approximation as ?
- How can we recognize an indeterminate form in limits?
- What other methods could we use to evaluate limits besides L'Hôpital's Rule?
- How does L'Hôpital's Rule apply to cases involving infinity?
Tip: When applying L'Hôpital's Rule, make sure to check that the limit is indeed an indeterminate form (such as or ) before proceeding with differentiation.
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Math Problem Analysis
Mathematical Concepts
Limits
L'Hôpital's Rule
Trigonometric Functions
Logarithmic Functions
Formulas
L'Hôpital's Rule: \( \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \)
Approximation of \( \sin(x) \approx x \) as \( x \to 0 \)
Theorems
L'Hôpital's Rule
Suitable Grade Level
College Level Calculus
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