How can I find the following limit: $\lim\limits_{x\to \:4}\frac{\sqrt{x+5}-3}{x-4}$

Question

Find the following limit:$$\lim\limits_{x\to \:4}\frac{\sqrt{x+5}-3}{x-4}$$

I tried to multiply by the conjugate and it did not work.

Answer 1

If you multiply by the conjugate, you should find

$$\frac{(x + 5) - 9}{(x - 4)(\sqrt{x + 5} + 3)} = \frac{x - 4}{(x - 4)\sqrt{x + 5} + 3}$$

Can you finish?

Answer 2

In case you are familiar with derivatives: this limit is the derivative of $f(x)=\sqrt{x+5}$ at $x=4$.