Question

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

Answer 1

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.

Answer 2

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