Question

How do you find the limit of `(e^x + x)^(1/X)` as x approaches infinity?

Answer 1

`e`

Explanation:

As `x` gets very big the exponential term will completely dominate the linear term, so with that in mind we can conclude that as `x` gets very big the expression becomes:

`(e^x+x)^(1/x)->(e^x)^(1/x)` as `x->oo`

`(e^x)^(1/x)=e^(x/x)=e^1=e`

So the limit is `e`.

If we look at the graph we can see an asymptote at `e` as we would expect:

graph{(e^x+x)^(1/x) [-2.87, 17.13, -1.78, 8.22]}