Evaluate $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{e^x-1-x}{x^2}}$

The limit of natural exponential function in $x$ minus one minus $x$ divided by $x$ square should be evaluated in this limit problem as the value of $x$ approaches zero. Firstly, let us try to find the limit of rational function by the direct substitution.

$=\,\,$ $\dfrac{e^0-1-0}{0^2}$

According to the zero power rule, the mathematical constant $e$ raised to the power of zero is one.

$=\,\,$ $\dfrac{1-1-0}{0}$

$=\,\,$ $\dfrac{1-1}{0}$

$=\,\,$ $\dfrac{0}{0}$

According to the direct substitution, the limit $e$ raised to the power of $x$ minus $1$ minus $x$ divided by square of $x$ is indeterminate. So, we must think about other methods to find its limit. The limit of the given rational function can be evaluated in the following methods possibly.

L’Hôpital’s Rule

Evaluate $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{e^x-1-x}{x^2}}$

Learn how to find the limit of $e$ raised to the power of $x$ minus $1$ minus $x$ divided by $x$ square by the l’hospital’s rule.