Given that \lim_{x \rightarrow +\infty}(1 + 1/x)^{x }= e Show that \lim_{x \rightarrow +\infty}(1...


Given that {eq}\displaystyle \lim_{x \rightarrow +\infty}(1 + 1/x)^{x }= e . {/eq} Show that {eq}\displaystyle \lim_{x \rightarrow +\infty}(1 + k/x)^{x} = e^{k}. {/eq}


We have been given a standard limit formula and we have to use it to prove the given limit. We will do an arithmetic operation and then apply the laws of the exponent to prove the equation.

Answer and Explanation:

$$\lim_{x \rightarrow +\infty}(1 + k/x)^{x} \\ $$

We can also write it as:

$$\lim_{x\rightarrow \infty}(1+\frac{k}{x})^{\frac{x}{k}k}\\ \lim_{\frac{x}{k}\rightarrow 0}(1+\frac{k}{x})^{\frac{x}{k}k}\\ $$

By the given limit formula we can write it as:

$$e^{k} $$

Learn more about this topic:

Understanding the Properties of Limits

from Math 104: Calculus

Chapter 6 / Lesson 5

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