Question

Find the following limits:$\underset{x\to 3}{lim}\frac{\sqrt{4-x}-\sqrt{x-2}}{15-5x}$

Answer 1

given equation is

$\underset{x\to 3}{lim}\frac{\sqrt{4-x}-\sqrt{x-2}}{15-5x}$

multiplying and dividing it with $\sqrt{4-x}+\sqrt{x-2}$ we get

$\underset{x\to 3}{lim}\frac{(\sqrt{4-x}-\sqrt{x-2})(\sqrt{4-x}+\sqrt{x-2})}{(15-5x)(\sqrt{4-x}+\sqrt{x-2})}=\underset{x\to 3}{lim}\frac{\left(\right(4-x)-(x-2\left)\right)}{(15-5x)(\sqrt{4-x}+\sqrt{x-2})}$[$(a-b)(a+b)=$$a^2-b^2$]

$\underset{x\to 3}{lim}\frac{2(3-x)}{5(3-x)(\sqrt{4-x}+\sqrt{x-2})}=\underset{x\to 3}{lim}\frac{2}{5(\sqrt{4-x}+\sqrt{x-2})}\Rightarrow =\frac{2}{5(1+1)}=\frac{1}{5}$