Question

Question #62181

Answer 1

`lim_(x->0) (1-sqrt(1-x^2))/(3x)=`

Explanation:

We want to find `lim_(x->0) (1-sqrt(1-x^2))/(3x)`

First we plug in the limiting value to see if our limit is indeterminate or not.

`(1-sqrt(1-0^2))/(3(0))=0/0` so the limit is finite.

`lim_(x->0) (1-sqrt(1-x^2))/(3x)=lim_(x->0)((1-sqrt(1-x^2))(1+sqrt(1-x^2)))/((3x)(1+sqrt(1-x^2)))=lim_(x->0)(1-(1-x^2))/((3x)(1+sqrt(1-x^2)))=lim_(x->0)x^2/((3x)(1+sqrt(1-x^2)))=lim_(x->0) x/(3(1+sqrt(1-x^2))`

Now we see that plugging in `0` won't give us an expression which we can't evaluate so we can plug in `0` again

The limit is `0/(3(1+sqrt(1-0^2)))=0`