Math Problem Statement

evaluate

Evaluate lim(x→0) (1 + sin(x))^cot(x)

Solution

The given limit is:

limx0(1+sinx)cotx\lim_{{x \to 0}} \left(1 + \sin x\right)^{\cot x}

To evaluate this limit, we can use the exponential form to handle the indeterminate expression. Let's rewrite it as:

limx0(1+sinx)cotx=elimx0cotxln(1+sinx)\lim_{{x \to 0}} \left(1 + \sin x\right)^{\cot x} = e^{\lim_{{x \to 0}} \cot x \cdot \ln(1 + \sin x)}

Now, we will evaluate the inner limit:

limx0cotxln(1+sinx)\lim_{{x \to 0}} \cot x \cdot \ln(1 + \sin x)

We know that cotx=cosxsinx\cot x = \frac{\cos x}{\sin x} and, as x0x \to 0, sinxx\sin x \approx x and cosx1\cos x \approx 1, so cotx1x\cot x \approx \frac{1}{x}. Also, for small values of xx, ln(1+sinx)sinx=x\ln(1 + \sin x) \approx \sin x = x.

Thus, the limit simplifies to:

limx0ln(1+sinx)sinxlimx0xx=1\lim_{{x \to 0}} \frac{\ln(1 + \sin x)}{\sin x} \approx \lim_{{x \to 0}} \frac{x}{x} = 1

Therefore, the exponential form of the limit is:

elimx0cotxln(1+sinx)=e1=ee^{\lim_{{x \to 0}} \cot x \cdot \ln(1 + \sin x)} = e^{1} = e

The final answer is:

limx0(1+sinx)cotx=e\lim_{{x \to 0}} \left(1 + \sin x\right)^{\cot x} = e

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

  1. How would you evaluate the limit of (1+sinx)1/x(1 + \sin x)^{1/x} as x0x \to 0?
  2. What is the general approach to solving limits involving exponential forms?
  3. How does L'Hospital's Rule apply to limits like this?
  4. How do you determine when a limit tends towards ee in indeterminate forms?
  5. What are the conditions under which limx0(1+x)1x=e\lim_{{x \to 0}} \left(1 + x\right)^{\frac{1}{x}} = e?

Tip: For limits involving forms like 000^0, 11^\infty, or 0\infty^0, 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