# Use Continuity to evaluate the limit, $$\lim_{x \to 4} \left ( \frac{5 + \sqrt{x}}{\sqrt{5 + x}} \right )$$

Solution:

Given

$$\lim_{x \to 4} \left ( \frac{5 + \sqrt{x}}{\sqrt{5 + x}} \right )$$

Clearly $$\sqrt{x}$$ is continuous for $$x \geq 0$$ and $$\sqrt{(5+x)}$$ is continuous for

$$(5 + x) \geq 0 \enspace \Rightarrow \enspace x \geq -5$$

Therefore,

$$\frac{5 + \sqrt{x}}{\sqrt{5 + x}}$$             … (ii)

is continuous for $$x \geq 0$$. So, Limit of (ii) exists. And,

$$= \lim_{x \to 4} \left ( \frac{5 + \sqrt{x}}{\sqrt{5 + x}} \right )$$

$$= \frac{5 + \sqrt{x}}{\sqrt{5 + x}}$$

$$= \frac{5 + 2}{3} = \frac{7}{3}$$