Q 78. - Calculus

Question

Find the limit.

$\underset{x\to 0}{\lim }{\left(1+2x\right)}^{\frac{3}{x}}$

Step-by-Step Solution

Answer

Answer is $e^6$

Step 1. Given information

Limit function is $\underset{x\to 0}{\lim }{\left(1+2x\right)}^{\frac{3}{x}}$

Step 2. Explanation

Take $2x=h$

$\begin{array}{rcl}\underset{x\to 0}{\lim }{\left(1+2x\right)}^{\frac{3}{x}}& =& \underset{h\to 0}{\lim }{\left(1+h\right)}^{\frac{6}{h}}\\ & =& {\left[\underset{h\to 0}{\lim }{\left(1+h\right)}^{\frac{1}{h}}\right]}^6\\ & =& e^6\end{array}$