Math Problem Statement
Solution
We are tasked with evaluating the limit:
Step 1: Substitute
First, let’s substitute directly into the expression:
So, at , the expression becomes:
This results in an indeterminate form , so we need to apply L'Hopital's Rule, which allows us to differentiate the numerator and denominator separately and then evaluate the limit.
Step 2: Apply L'Hopital's Rule
We need to differentiate the numerator and the denominator:
-
The numerator is . The derivative of is 0, and the derivative of is . So, the derivative of the numerator is:
-
The denominator is . The derivative of is , and the derivative of is . So, the derivative of the denominator is:
Step 3: Evaluate the new limit
Now we evaluate the new limit:
So, the limit becomes:
Final Answer:
Would you like more details on any step? Here are some related questions you might find useful:
- How do you apply L'Hopital's Rule in more complicated cases?
- What are the conditions under which L'Hopital’s Rule can be used?
- Can limits be solved using series expansions instead of L'Hopital’s Rule?
- How do you calculate derivatives of trigonometric functions like ?
- What is the geometric interpretation of the limit problem?
Tip: Always check the form of your limit before applying L'Hopital’s Rule to ensure it’s indeterminate, such as or .
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Math Problem Analysis
Mathematical Concepts
Limits
Trigonometry
L'Hopital's Rule
Derivatives
Formulas
lim (x → c) f(x) / g(x) = lim (x → c) f'(x) / g'(x) (L'Hopital's Rule)
Derivative of tan(x) is sec^2(x)
sin(π/4) = cos(π/4) = √2/2
sec^2(π/4) = 2
Theorems
L'Hopital's Rule
Suitable Grade Level
College-level Calculus
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