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Chapter 4: Problem 59

Evaluate the following limits or explain why they do not exist. Check your results by graphing. $$\lim _{x \rightarrow 0^{+}}(1+x)^{\cot x}$$

Short Answer

Expert verified
Answer: The limit as x approaches 0 from the positive side of the function \((1+x)^{\cot x}\) is \(e\).

Step by step solution

01

Rewrite the function using the cotangent identity

We start by rewriting the function using the cotangent identity: $$\cot x =\frac{\cos x}{\sin x}$$ So, the function becomes: $$(1+x)^{\frac{\cos x}{\sin x}}$$ Now, we'll find the limit as \(x\) approaches \(0\) from the positive side.
02

Natural logarithm trick to solve the limit

To find the limit of this function, we'll use the following natural logarithm trick: $$\lim_{x \rightarrow 0^+} (1+x)^{\frac{\cos x}{\sin x}} = e^{\lim_{x \rightarrow 0^+} \left( \frac{\cos x}{\sin x}\cdot \ln(1+x) \right)}$$ Now, we need to find the limit of the exponent: $$\lim_{x \rightarrow 0^+} \left( \frac{\cos x}{\sin x}\cdot \ln(1+x) \right)$$ At this point, we can use L'H么pital's rule to find the limit.
03

Apply L'H么pital's rule

We have the form \(\frac{0}{0}\) when \(x \rightarrow 0^+\). So, we apply L'H么pital's rule by differentiate the numerator and the denominator: $$\lim_{x \rightarrow 0^+} \frac{\frac{-\sin x \cdot \ln(1+x) + \cos x \cdot \frac{1}{1+x}}{\cos x}$$ Now, evaluate the limit as \(x \rightarrow 0^+\). $$ \frac{-\sin (0) \cdot \ln(1) + \cos (0) \cdot \frac{1}{1}}{\cos (0)} = \frac{1}{1} = 1$$ So, the limit of the exponent is 1.
04

Compute the final limit

Now that we have found the limit of the exponent, we can plug it back into the original limit function: $$\lim_{x \rightarrow 0^+} (1+x)^{\frac{\cos x}{\sin x}} = e^1 = e$$ So, the given limit is equal to \(e\).
05

Check the result graphically

To confirm our result, we can graph the function \((1+x)^{\frac{\cos x}{\sin x}}\) for \(x\) approaching \(0\) from the positive side. Notice that the graph approaches the value \(e\) as we get closer to \(0\) from the positive side, supporting our analysis and conclusion. As a result, the given limit as \(x\) approaches \(0\) from the positive side is \(e\).

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