Question

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

`lim_(x to oo) x/e^x`

Answer 1

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.

Answer 2

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`