How do you find the limit of #(e^x - 1)/x^3# as x approaches 0?

2 Answers
Mar 10, 2016

The limit does not exist, because, as #xrarr0# the function increases without bound. #lim_(xrarr0)(e^x-1)/x^3 = oo#

Explanation:

For his one, I would use l'Hospital's rule.

The initial forms of #lim_(xrarr0)(e^x-1)/x^3# is #0/0#.

Applying l'Hospital, gets us to

#lim_(xrarr0)e^x/(3x^2)# which has the form #1/0#

Because both the numerator and denominator are positive for #x# near #0#, this form tells us that the function increases without bound ad #xrarr0# from either side.

Mar 10, 2016

Infinity.

Explanation:

Differentiate numerator and denominator and take the limit for the ratio applying L' Hospital's rule.for limits of indeterminate forms.
The ratio of the derivatives is #e^x/(3x^2)#.
#e^0# = 1