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Factorization

lim x → 21+4 ...

Question

$\lim_{x \to 2} \frac{\sqrt{1 + 4 x} - \sqrt{5 + 2 x}}{x - 2}$

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Solution

$\lim_{x \to 2} [\frac{\sqrt{1 + 4 x} - \sqrt{5 + 2 x}}{x - 2}]$

It is of the from $\frac{0}{0}$ .
Rationalising the numerator:

$\lim_{x \to 2} [\frac{(\sqrt{1 + 4 x} - \sqrt{5 + 2 x}) (\sqrt{1 + 4 x} + \sqrt{5 + 2 x})}{(x - 2) (\sqrt{1 + 4 x} + \sqrt{5 + 2 x})}]$

= $\lim_{x \to 2} [\frac{(1 + 4 x) - (5 + 2 x)}{(x - 2) (\sqrt{1 + 4 x} + \sqrt{5 + 2 x})}]$

= $\lim_{x \to 2} [\frac{2 (x - 2)}{(x - 2) (\sqrt{1 + 4 x} + \sqrt{5 + 2 x})}]$

= $\frac{2}{(\sqrt{1 + 4 \times 2} + \sqrt{5 + 2 \times 2})}$

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

= $\frac{1}{3}$

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