Q 80. - Calculus

Question

Find the limit.

$\underset{x\to \infty }{\lim }{\left(1-\frac{5}{x}\right)}^x$

Step-by-Step Solution

Answer

Answer is $\frac{1}{e^5}$

Step 1. Given information

Limit function is $\underset{x\to \infty }{\lim }{\left(1-\frac{5}{x}\right)}^x$

Step 2. Explanation

Let $\frac{5}{x}=-h$

$\begin{array}{rcl}\underset{x\to \infty }{\lim }{\left(1-\frac{5}{x}\right)}^x& =& \underset{h\to 0}{\lim }{\left(1+h\right)}^{\frac{-5}{h}}\\ & =& \underset{h\to 0}{\lim }\left[\frac{1}{{\left(1+h\right)}^{{\displaystyle \frac{5}{h}}}}\right]\\ & =& \frac{1}{e^5}\end{array}$