Lim x - 2 x 2-1-5 x-2

Limx #8594;2x2+1 5x 2 It is of the form 00. Rationalising the numerator: nbsp;limx #8594;2x2+1 5x2+1+5x 2x2+1+5 = nbsp;limx #8594;2x2+1 5x 2x2+1+5 ...

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Byju's Answer

Standard XII

Mathematics

Special Integrals - 2

lim x → 2 x 2...

Question

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

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Solution

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

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

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

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

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

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

= $\frac{4}{2 \sqrt{5}}$

= $\frac{2}{\sqrt{5}}$

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limx→2 x2+1-5x-2

limx→2x2+1-5x-2 It is of the form 00. Rationalising the numerator: limx→2x2+1-5x2+1+5x-2x2+1+5 = limx→2x2+1-5x-2x2+1+5 = limx→2x2-4x-2x2+1+5 = limx→2x-2x+2x-2x2+1+5 = 425 = 25

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