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Chapter 4: Problem 61

Evaluate the following limits or explain why they do not exist. Check your results by graphing. $$\lim _{x \rightarrow \infty}\left(1+\frac{a}{x}\right)^{x}, \text { for a constant } a$$

Short Answer

Expert verified
Answer: The limit exists and is equal to $e^a$.

Step by step solution

01

Recognize the limit expression

The given limit expression resembles the definition of the exponential function \(e^a\), which is defined as: $$e^a = \lim_{n\rightarrow\infty}\left(1+\frac{a}{n}\right)^{n}.$$ Notice that here, we used \(n\) instead of \(x\) for consistency with the standard definition, but it doesn't change the essence of the problem. So, our goal is to evaluate the limit, knowing that it resembles the limit definition of the exponential function.
02

Evaluate the limit

Using the fact that the given limit resembles the definition of the exponential function \(e^a\), we can rewrite the limit as: $$\lim_{x \rightarrow \infty}\left(1+\frac{a}{x}\right)^{x} = e^a$$ That's because, as \(x\rightarrow\infty\), the expression \(\left(1+\frac{a}{x}\right)^{x}\) approaches the exponential function \(e^a\). So, the limit exists, and is equal to \(e^a\).
03

Check the result by graphing

In order to verify our result, we can graph the function \(f(x) = \left(1+\frac{a}{x}\right)^{x}\) for a constant value of \(a\). Regardless of the value of \(a\), as \(x\rightarrow\infty\), the graph of the function should approach \(e^a\). You can try this with a graphing software by choosing different values of \(a\) and observing the behavior of the function as \(x\) increases. For example, if we choose \(a=1\), the function would be \(f(x) = \left(1+\frac{1}{x}\right)^{x}\), and as \(x\rightarrow\infty\), the graph of this function should approach \(e^1 = e \approx 2.718\). The graph of the function should confirm this result.

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