Limit Evaluation when x reach 14

Question

$$\lim_{x\rightarrow 14}\frac{\sqrt{x-5}-3}{x-14}$$

How do I evaluate the limit when I put x = 14 and I got 0/0?

Answer 1

write $$\frac{\sqrt{x-5}-3}{x-14}\cdot \frac{\sqrt{x-5}+3}{\sqrt{x-5}+3}$$

Answer 2

$$\frac{\sqrt{x-5}-3}{x-14}=\frac{x-14}{(x-14)(\sqrt{x-5}+3)}\rightarrow\frac{1}{6}.$$

Answer 3

Hint:

$$x-14=(\sqrt{x-5}+3)(\sqrt{x-5}-3)$$

Answer 4

By L'Hôpital's rule

$$\lim_{x\rightarrow 14}\frac{\sqrt{x-5}-3}{x-14}=\lim_{x\rightarrow 14}\frac{\frac{1}{2}}{\sqrt{x-5}}=\frac{1}{6}$$