Question

Evaluate each of the following limits, if exists
i) $\underset{x\to -1}{lim}\frac{x^3-3x+1}{x-1}$
ii) $\underset{x\to a}{lim}\frac{\sqrt{a+2x}-\sqrt{3x}}{\sqrt{3a+x}-2\sqrt{x}},a\ne 0$

Answer 1

i)

${lim}_{x\to -1}\frac{x^3-3x+1}{x-1}$

$=\frac{(-1{)}^3-3(-1)+1}{-1-1}$

$=\frac{-1+3+1}{-2}$

$=\frac{-3}{2}$

ii)

${lim}_{x\to a}\frac{\sqrt{a+2x}-\sqrt{3x}}{\sqrt{3a+x}-2\sqrt{x}}$

Differentiate w.r.t x

${lim}_{x\to a}\frac{\frac{1}{\sqrt{a+2x}}-\frac{\sqrt{3}}{2\sqrt{x}}}{\frac{1}{2\sqrt{3a+x}}-\frac{1}{\sqrt{x}}}$

${lim}_{x\to a}\frac{\sqrt{3a+x}(2\sqrt{x}-\sqrt{3a+6x})}{\sqrt{a+2x}(\sqrt{x}-2\sqrt{3a+x})}$

$=\frac{2\sqrt{3}}{9}$