Question

What is the limit of `x^(1-x)` as x approaches 1? The hint is "Use the properties of logarithms to simplify the limit". But I just can think in the straight answer 1, by substitution. Can someone develop in steps using this hint, please?

Answer 1

The limit is `1`, as you suspected.

Explanation:

It is true you can use substitution but I will show you the logarithm method so that you know how to do it when substitution doesn't work.

`L = lim_(x-> 1) x^(1- x)`

`lnL = lim_(x-> 1) ln(x^(1 -x))`

`lnL = lim_(x-> 1) (1 -x)lnx`

`lnL = 0`

`L = e^0`

`L = 1`

Hopefully this helps!

Answer 2

A slightly different way but with a similar approach

Explanation:

As Noah showed, one way to find the limit is to take the natural log of both sides. However, a mistake students might make is that they forget to undo the log. So here's a slightly different way to do it.

`lim_(x->1)x^(1-x)=lim_(x->1)e^ln(x^(1-x))`

`color(white)(a)=e^ (lim_(x->1) (1-x)lnx)`

`color(white)(a)=e^0`

`color(white)(a)=1`

This way, we avoid having to undo anything at the end, and we can present our final answer, hopefully, without making any mistakes.