Evaluate lim x/1-e^x (Hint: Use L’Hosiptal’s Rule, but make sure it applies first)

Answer 1

You don't say which limit you're looking for - could be x → −∞, x → 0, x → ∞, for example. Let's look at those three:

1. As x → −∞, 1 - e^x approaches 1. −∞/1 is not an indeterminate form, so L'Hospital's rule does not apply.
2. As x → 0, 1 - e^x approaches 0, so x/(1 - e^x) takes the form 0/0. Here you can use L'Hospital: dx/dx = 1, dx (1 - e^x)/dx = - e^x. 1/(-e⁰) = -1.
3. As x → ∞, 1 - e^x approaches −∞. Using L'Hospital's rule leads to 1/(−∞).