# 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?

##### 2 Answers

The limit is

#### 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!

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.

#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.