Math Problem Statement
evaluate
Solution
The given limit is:
To evaluate this limit, we can use the exponential form to handle the indeterminate expression. Let's rewrite it as:
Now, we will evaluate the inner limit:
We know that and, as , and , so . Also, for small values of , .
Thus, the limit simplifies to:
Therefore, the exponential form of the limit is:
The final answer is:
Would you like more details or have any questions?
Here are 5 related questions for further exploration:
- How would you evaluate the limit of as ?
- What is the general approach to solving limits involving exponential forms?
- How does L'Hospital's Rule apply to limits like this?
- How do you determine when a limit tends towards in indeterminate forms?
- What are the conditions under which ?
Tip: For limits involving forms like , , or , rewriting the expression in exponential form is a helpful strategy.
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Math Problem Analysis
Mathematical Concepts
Limits
Exponential Functions
Trigonometric Limits
Formulas
lim(x→0) (1 + sin(x))^cot(x) = e^lim(x→0) [cot(x) * ln(1 + sin(x))]
cot(x) ≈ 1/x as x → 0
ln(1 + sin(x)) ≈ x as x → 0
Theorems
Exponential Limit Theorem
Taylor Expansion Approximation
Suitable Grade Level
Undergraduate Calculus