Lim x - 13-x-5-x x 2-1

Limx #8594;13+x 5 xx2 1 It is of the form 00. Rationalising the numerator: nbsp;limx #8594;13+x 5 x3+x+5 xx 1x+13+x+5 x = nbsp;limx #8594;13+x 5 x ...

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Byju's Answer

Standard XII

Mathematics

Factorization

lim x → 13+x-...

Question

$\lim_{x \to 1} \frac{\sqrt{3 + x} - \sqrt{5 - x}}{x^{2} - 1}$

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Solution

$\lim_{x \to 1} [\frac{\sqrt{3 + x} - \sqrt{5 - x}}{x^{2} - 1}]$

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

$\lim_{x \to 1} [\frac{(\sqrt{3 + x} - \sqrt{5 - x}) (\sqrt{3 + x} + \sqrt{5 - x})}{(x - 1) (x + 1) (\sqrt{3 + x} + \sqrt{5 - x})}]$

= $\lim_{x \to 1} [\frac{(3 + x) - (5 - x)}{(x - 1) (x + 1) \{\sqrt{3 + x} + \sqrt{5 - x}\}}]$

= $\lim_{x \to 1} [\frac{2 x - 2}{(x - 1) (x + 1) \{\sqrt{3 + x} + \sqrt{5 - x}\}}]$

= $\lim_{x \to 1} [\frac{2 (x - 1)}{(x - 1) (x + 1) \{\sqrt{3 + x} + \sqrt{5 - x}\}}]$

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

= $\frac{1}{4}$

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limx→1 3+x-5-xx2-1

limx→13+x-5-xx2-1 It is of the form 00. Rationalising the numerator: limx→13+x-5-x3+x+5-xx-1x+13+x+5-x = limx→13+x-5-xx-1x+13+x+5-x = limx→12x-2x-1x+1 3+x+5-x = limx→12x-1x-1x+13+x+5-x = 22 4+4 = 14

$\lim_{x \to 1} \frac{\sqrt{3 + x} - \sqrt{5 - x}}{x^{2} - 1}$