Question

Evaluate
$\rm \displaystyle \lim_{x\to 2}\ \frac{\sqrt{x^{3}-4}-2}{\sqrt{20-x^{2}}-4}$

Answer 1

For such type of limit expression we had to rationalized both numerator and denominator

$\lim_{x\to 2}\ \frac{\sqrt{x^{3}-4}-2}{\sqrt{20-x^{2}}-4} \\ \\ = \lim_{x\to 2}\ \frac{\sqrt{x^{3}-4}-2}{\sqrt{20-x^{2}}-4} \times \frac{ \sqrt{ {x}^{3} - 4} + 2}{ \sqrt{20 - {x}^{2} } + 4} \times \frac{\sqrt{20 - {x}^{2} } + 4}{\sqrt{ {x}^{3} - 4} + 2} \\ \\ = \lim_{x\to 2} \frac{( {x}^{3} - 4 - 4)}{20 - {x}^{2} - 16 } \times \frac{\sqrt{20 - {x}^{2} } + 4}{\sqrt{ {x}^{3} - 4} + 2} \\ \\ = \lim_{x\to 2} \frac{( {x}^{3} - 8)}{ (- {x}^{2} + 4) } \times \frac{\sqrt{20 - {x}^{2} } + 4}{\sqrt{ {x}^{3} - 4} + 2} \\ \\ apply \: identities \\ \\ {a}^{3} - {b}^{3} = (a - b)( {a}^{2} + ab + {b}^{2} ) \\ \\ {a}^{2} - {b}^{2} = (a + b)(a - b) \\ \\ = \lim_{x\to 2} \frac{( {x} - 2)( {x}^{2} + 2x + 4)}{ - ( {x} + 2) (x - 2)} \times \frac{\sqrt{20 - {x}^{2} } + 4}{\sqrt{ {x}^{3} - 4} + 2} \\ \\ = \lim_{x\to 2} \frac{( {x}^{2} + 2x + 4)}{ - ( {x} + 2)} \times \frac{\sqrt{20 - {x}^{2} } + 4}{\sqrt{ {x}^{3} - 4} + 2} \\ \\now \: apply \: limit \\ \\ \frac{({(2)}^{2} + 2(2) + 4)}{ - ( {2} + 2)} \times \frac{\sqrt{20 - {(2)}^{2} } + 4}{\sqrt{ {(2)}^{3} - 4} + 2} \\ \\ = \frac{4 + 4 + 4}{ - 4} \times \frac{ \sqrt{16} + 4 }{ \sqrt{4} + 2} \\ \\ = \frac{ - 12}{4} \times \frac{8}{4} \\ \\\lim_{x\to 2}\ \frac{\sqrt{x^{3}-4}-2}{\sqrt{20-x^{2}}-4} = - 6$
Hope it helps you.