Scientific Computing - Quiz 1 - Question 3

Problem:

L'Hopital's rule is a method for determining the value of indeterminate forms. Determine the value of the following limits
$\lim\limits_{x \to 0} \frac{e^x - 1 - x}{x^2}$

Solution:

$\begin{eqnarray*} & & \lim\limits_{x \to 0} \frac{e^x - 1 - x}{x^2} \\ &=& \lim\limits_{x \to 0} \frac{e^x - 1}{2x} \\ &=& \lim\limits_{x \to 0} \frac{e^x}{2} \\ &=& \frac{1}{2} \\ \end{eqnarray*}$

Again, this is easy to validate numerically:

x	f(x)	$e^2$
0.1	0.517091808	0.000292
0.01	0.501670842	2.79E-06
0.001	0.500166708	2.78E-08
0.0001	0.500016671	2.78E-10
0.00001	0.500000696	4.85E-13