Lim_x → 1x 1/√(x^2+3) 2 Rationalising the denominator =lim_x → 1(x 1)× (√(x^2+3)+2)/(√(x^2+3) 2)(√(x^2+3)+2) =lim_x → 1(x 1)× (√(x^2+3)+2)/(x^2+3 4) =lim_x → 1( ...

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Byju's Answer

Standard XII

Mathematics

Rationalization Method to Remove Indeterminate Form

lim x → 1x-1/...

Question

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

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Solution

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

Rationalising the denominator

$= {lim}_{x \to 1} \frac{(x - 1) \times (\sqrt{x^{2} + 3} + 2)}{(\sqrt{x^{2} + 3} - 2) (\sqrt{x^{2} + 3} + 2)}$

$= {lim}_{x \to 1} \frac{(x - 1) \times (\sqrt{x^{2} + 3} + 2)}{(x^{2} + 3 - 4)}$

$= {lim}_{x \to 1} \frac{(x - 1) \times (\sqrt{x^{2} + 3} + 2)}{(x^{2} - 1)}$

$= {lim}_{x \to 1} \frac{(x - 1) \times (\sqrt{x^{2} + 3} + 2)}{(x - 1) (x + 1)}$

$= {lim}_{x \to 1} \frac{\sqrt{x^{2} + 3} + 2}{x + 1}$

$\Rightarrow \frac{\sqrt{1 + 3} + 2}{1 + 1}$

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

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

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is equal to?

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limx→1x−1√x2+3−2

limx→1x−1√x2+3−2 Rationalising the denominator =limx→1(x−1)×(√x2+3+2)(√x2+3−2)(√x2+3+2) =limx→1(x−1)×(√x2+3+2)(x2+3−4) =limx→1(x−1)×(√x2+3+2)(x2−1) =limx→1(x−1)×(√x2+3+2)(x−1)(x+1) =limx→1√x2+3+2x+1 ⇒√1+3+21+1 =2+22 =42=2

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