Q 79. - Calculus

Question

Find the limit.

$\underset{x\to \infty }{\lim }{\left(1+\frac{1}{x}\right)}^{3x}$

Step-by-Step Solution

Answer

Answer is $e^3$

Step 1. Given information

Limit function is $\underset{x\to \infty }{\lim }{\left(1+\frac{1}{x}\right)}^{3x}$

Step 2. Explanation

Let

$\begin{array}{rcl}\underset{x\to \infty }{\lim }{\left(1+\frac{1}{x}\right)}^{3x}& =& \underset{h\to 0}{\lim }{\left(1+h\right)}^{\frac{3}{h}}\\ & =& {\left[\underset{h\to 0}{\lim }{\left(1+h\right)}^{\frac{1}{h}}\right]}^3\\ & =& e^3\end{array}$