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Chapter 11: Problem 26

Calculate. $$\lim _{x \rightarrow \infty}\left(1+\frac{a}{x}\right)^{b x}$$

Short Answer

Expert verified
The limit as \(x\) approaches infinity for \(\left(1 + \frac{a}{x}\right)^{b x}\) is \(e^{ab}\).

Step by step solution

01

Recall the limit definition of Euler's Number \(e\)

The mathematical constant \(e\) can be represented as a limit in the form: \(e = \lim_{n \rightarrow \infty} \left(1 + \frac{1}{n}\right)^n\). This definition will be essential for the next step.
02

Match the given exercise to the limit definition of \(e\)

Comparing the given limit \(\lim_{x \rightarrow \infty} \left(1 + \frac{a}{x}\right)^{b x}\) with the limit definition of \(e = \lim_{n \rightarrow \infty} \left(1 + \frac{1}{n}\right)^n\), we observe that if we choose \(a = 1\) and \(b = n\), our problem matches the definition of \(e\). We will use this observation to rewrite our limit.
03

Rewrite the given limit using properties of exponentiation

By using the rule \((a^m)^n = a^{mn}\), we can rewrite the expression within the limit as power of \(e\). We get: \(\lim_{x \rightarrow \infty} \left[\left(1 + \frac{a}{x}\right)^x\right]^{b}\). By the observation made in step 2, we know that \(\lim_{x \rightarrow \infty} \left(1 + \frac{1}{x}\right)^x = e^a\). Thus, our rewritten limit becomes \(e^{ab}\).
04

Evaluate the limit

Since \(e^{ab}\) is a constant expression, the limit as \(x\) approaches infinity is simply the constant itself. Thus, \(\lim_{x \rightarrow \infty} \left(1 + \frac{a}{x}\right)^{b x} = e^{ab}\).

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