Question

What is the limit of `(1+2x)^(1/x)` as x approaches infinity?

Answer 1

`lim_(x->oo) (1+2x)^(1/x) = 1`

Explanation:

Write the function as:

`(1+2x)^(1/x) = (e^(ln(1+2x)))^(1/x) = e^(ln(1+2x)/x)`

Now evaluate:

`lim_(x->oo) ln(1+2x)/x`

This limit is in the indeterminate form `oo/oo` so we can solve it using l'Hospital's rule:

`lim_(x->oo) ln(1+2x)/x = lim_(x->oo) (d/dx ln(1+2x))/(d/dx x) = lim_(x->oo) 2/(1+2x) = 0`

As `e^x` is a continuous function we have then:

`lim_(x->oo) (1+2x)^(1/x) = lim_(x->oo) e^(ln(1+2x)/x) = e^((lim_(x->oo) ln(1+2x)/x)) = e^0 = 1`