Evaluate the limit((Sqrt(4+x))-2)/(x)

1 Answer | Add Yours

thilina-g's profile pic

thilina-g | College Teacher | (Level 1) Educator

Posted on

You have not included the limit. But by looking at the question I can see that the intended limit is 0.

`lim_(x-gt0)(sqrt(4+x)-2)/x`

If you try to evaluate the limit straight away, you will see that it will result in 0/0 situation which is indertminate. To remove this, we can multiply both numerator and denominator by `(sqrt(4+x)+2)` .

`lim_(x-gt0)((sqrt(4+x)-2)(sqrt(4+x)+2))/(x(sqrt(4+x)+2))`

This would give,

`lim_(x-gt0)((4+x)-2^2)/(x(sqrt(4+x)+2))`

`lim_(x-gt0)x/(x(sqrt(4+x)+2))`

This simplifies into,

`lim_(x-gt0)1/(sqrt(4+x)+2)`

Now you can evaluate the limit easily,

`lim_(x-gt0)1/(sqrt(4+x)+2) = 1/(2+2) = 1/4`

 

Therefore,

`lim_(x-gt0)(sqrt(4+x)-2)/x = 1/4`

 

We’ve answered 315,489 questions. We can answer yours, too.

Ask a question