Match the limx-fx in col- 1 with col- 2

A) lim _ x→∞ #160; f( x ) #160; =lim _ x→∞ #160;√( (x+1)^ 2 )√( (x 1)^ 2 ) #160; =lim _ x→∞ #160; (x+1)^ 2 (x 1)^ 2 (x+1)^ 4/3 +(x 1)^ 4/3 + ...

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Byju's Answer

Standard XII

Mathematics

Rationalization Method to Remove Indeterminate Form

Match the l...

Question

Match the ${lim}_{x \to \infty} f (x)$ in col. 1 with col. 2

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Solution

A) ${lim}_{x \to \infty} f (x) = {lim}_{x \to \infty} \sqrt[3]{(x + 1)^{2} -} \sqrt[3]{(x - 1)^{2}} = {lim}_{x \to \infty} \frac{(x + 1)^{2} - (x - 1)^{2}}{(x + 1)^{4 / 3} + (x - 1)^{4 / 3} + (x^{2} - 1)^{2 / 3}}$
$= {lim}_{x \to \infty} \frac{4 x^{- 1 / 3}}{(1 + x^{- 1})^{4 / 3} + (1 - x^{- 1})^{4 / 3} + (1 - x^{- 2})^{2 / 3}} = \frac{0}{1 + 1 + 1} = 0$
B) ${lim}_{x \to \infty} f (x) = {lim}_{x \to \infty} x^{3 / 2} (\sqrt{x^{3} + 1} - \sqrt{x^{3} - 1}) = {lim}_{x \to \infty} \frac{2 x^{3 / 2}}{\sqrt{x^{3} + 1} + \sqrt{x^{3} - 1}}$
$= {lim}_{x \to \infty} \frac{2 x^{3 / 2}}{1 - \sqrt{x^{- 3}} + \sqrt{1 - x^{- 3}}} = 1$
C) ${lim}_{x \to \infty} f (x) = {lim}_{x \to \infty} (\sqrt{x^{2} - 2 x - 1} - \sqrt{x^{2} - 7 x + 3}) = {lim}_{x \to \infty} \frac{5 x - 4}{\sqrt{x^{2} - 2 x - 1} - \sqrt{x^{2} - 7 x + 3}}$
$= {lim}_{x \to \infty} \frac{5 - 4 / x}{\sqrt{1 - 2 / x - 1} + \sqrt{1 - 7 / x + 3 / x^{2}}} = \frac{5}{2}$
D) ${lim}_{x \to \infty} f (x) = {lim}_{x \to \infty} \sqrt{(x + 1) (x + 2)} - x = {lim}_{x \to \infty} \frac{3 x + 2}{\sqrt{(x + 1) (x + 2)} + x}$
$= {lim}_{x \to \infty} \frac{3 + 2 / x}{\sqrt{(1 + x^{- 1}) (1 + 2 x^{- 1}) + 1}} = \frac{3}{2}$

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Match the limx→∞f(x) in col. 1 with col. 2

Match the ${lim}_{x \to \infty} f (x)$ in col. 1 with col. 2