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Chapter 4: Problem 48

Evaluate the following limits. $$\lim _{x \rightarrow \infty}\left(\csc (1 / x)\left(e^{1 / x}-1\right)\right)$$

Short Answer

Expert verified
Question: Evaluate the limit of the given expression: $$\lim _{x \rightarrow \infty}\left(\csc (1 / x)\left(e^{1 /x}-1\right)\right)$$ Answer: The limit of the given expression is 1.

Step by step solution

01

Rewrite the limit as the product of two functions

First, rewrite the given limit as the product of two separate limits: $$\lim _{x \rightarrow \infty}\left(\csc (1 / x)\left(e^{1 /x}-1\right)\right) = \lim _{x \rightarrow \infty} \csc(1/x) \cdot \lim_{x \rightarrow \infty} \left(e^{1/x} -1\right)$$
02

Apply L'Hopital's Rule for indeterminate products

The current limit is the indeterminate form of the product 0 * 鈭�, so we will apply L'Hopital's Rule for indeterminate products. Rewrite the limit as a single function: $$\lim _{x \rightarrow \infty} \frac{\left(e^{1 /x}-1\right)}{\sin(1/x)}$$
03

Apply L'Hopital's Rule

Differentiate both the numerator and denominator with respect to x, and apply L'Hopital's Rule: $$\lim _{x \rightarrow \infty} \frac{\frac{-1}{x^{2}}e^{1/x} }{-\frac{1}{x^{2}}\cos(1/x)} = \frac{\mbox{-}1}{1} \cdot \lim _{x \rightarrow \infty} \frac{e^{1/x}}{\cos(1/x)}$$ As x approaches infinity, the fraction exponent approaches 0, so we have: $$\lim _{x \rightarrow \infty} \frac{1}{\cos(1/x)}$$
04

Evaluate the limit

As x approaches infinity, the cosine function will have values between -1 and 1, so the limit becomes: $$\frac{1}{1} = 1$$ Thus, the limit of the given expression is 1.

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