How to solve limit x/e^x without using l'hospital rule?

lim_(x to oo) x/e^x

2 Answers
Feb 24, 2018

Due to knowledge of linear functions and exponential functions, we can say that lim_(x->∞)x/e^x=0

Explanation:

lim_(x->∞)x/e^x=∞/e^∞=∞/∞

This is an indeterminate form, but it tells us that as x->∞, both x and e^x ->∞.

The final answer will depend on which function goes toward faster.

Exponential functions always grow faster than linear functions; evaluating e^x and x at a few increasingly large points or comparing their graphs will tell us this. This means that e^x->∞ far faster than x->∞. Since the denominator is growing faster than the numerator, the limit will end up being 0.

Feb 24, 2018

We know:

e^x>x for all x inRR

The difference between the functions also increases as x increases (you may need to use derivatives to prove this, but an intuition is that if x increases by 1, e^x increases by e~~2.71).

This means that as x->oo, the numerator will become insignificant, and e^x will dominate. This means that our limit will tend to 0:

lim_(x->oo) x/e^x=0