Question

What is the limit of `(sqrt(x+5) - 3)/(x - 4)` as x approaches 4?

Answer 1

Let's multiply both numerator and denominator of this expression by `sqrt(x+5)+3` to get rid of undefined `0/0` value.

Assuming `x!=4` in our expression,

`(sqrt(x+5)-3) / (x-4) =`

`= [(sqrt(x+5)-3) * (sqrt(x+5)+3)] / [(x-4)*(sqrt(x+5)+3)] =`

`= (x+5-9) / [(x-4)*(sqrt(x+5)+3)] =`

`= (x-4)/ [(x-4)*(sqrt(x+5)+3)] =`

`= 1/ (sqrt(x+5)+3)`

As `x->4`, this expression tends to `1/(sqrt(4+5)+3)=1/6`.

Therefore,

`lim_(x->4)(sqrt(x+5)-3) / (x-4) = 1/6`

Answer 2

`frac{1}{6}`

Explanation:

`lim_{x->4}(sqrt{x+5}-3)=0`

`lim_{x->4}(x-4)=0`

Using the L'Hospital Rule,

`lim_{x->4}frac{sqrt{x+5}-3}{x-4}`

`=lim_{x->4}frac{frac{d}{dx}(sqrt{x+5}-3)}{frac{d}{dx}(x-4)}`

`=lim_{x->4}frac{(frac{1}{2sqrt{x+5}})}{1}`

`=frac{1}{2sqrt{4+5}}`

`=1/6`