Question

What is the limit of `x/e^x` as x approaches infinity?

Answer 1

`lim_(x->∞) x/e^x = 0`

Explanation:

Since direct substitution yields an indeterminate form of `∞/∞`, we can conclude that we can apply L'Hospital's rule, which basically involves taking a derivative of the numerator and the denominator.

In our case, we have

`lim_(x->∞) x/e^x = lim_(x->∞) 1/e^x = 0`

This makes sense because if we look at the graph of the original function, we can see that the function clearly approaches `0` as `x->∞`.

graph{x/e^x [-1.856, 5.072, -1.588, 1.877]}