Question

Find the limit. use l'hospital's rule if appropriate. if there is a more elementary method, consider using it. lim x→0 e3x − 1 − 3x x2

Answer 1

It looks like the limit is

$\displaystyle \lim_{x\to0} \frac{e^{3x} - 1 - 3x}{x^2}$

L'Hôpital's rule works in this case; applying it twice gives

$\displaystyle \lim_{x\to0} \frac{e^{3x} - 1 - 3x}{x^2} = \lim_{x\to0} \frac{3e^{3x} - 3}{2x} = \lim_{x\to0} \frac{9e^{3x}}{2} = \boxed{\frac92}$