Q. 28 - Calculus

Question

Find the limit by hand.

$\underset{x\to 0}{\lim } \frac{x^2}{e^x-1}$

Step-by-Step Solution

Answer

On solving the limit, we get, $0$.

Step 1. Given information.

Consider the given question,

$\underset{x\to 0}{\lim } \frac{x^2}{e^x-1}$

Step 2. Solve the given limit.

Rewriting the expression,

$=\underset{x\to 0}{\lim } \left[\left(\frac{x}{e^x-1}\right)x\right]$

Using the product rule,

$=\underset{x\to 0}{\lim } \frac{x}{e^x-1} \underset{x\to 0}{\lim } x$

As $\underset{h\to 0}{\lim } \frac{h}{e^h-1}=1$,

$$\begin{gathered} =1\left(0\right) \\ =0 \end{gathered}$$