Lim x - 0-1-x-1-x-2 x

Lim_x → 0√(1+x) √(1 x)/2x Rationalising the numerator =lim_x → 0(√(a+x) √(1 x))/2x×(√(1+x) √(1 x))/(√(1+x)+√(1 x)) =lim_x → 0(1+x) (1 x)/(√(1+x)+√(1 x)) =lim_x ...

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Byju's Answer

Standard XII

Mathematics

Rationalization Method to Remove Indeterminate Form

lim x → 0√1+x...

Question

${lim}_{x \to 0} \frac{\sqrt{1 + x} - \sqrt{1 - x}}{2 x}$

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Solution

${lim}_{x \to 0} \frac{\sqrt{1 + x} - \sqrt{1 - x}}{2 x}$

Rationalising the numerator

$= {lim}_{x \to 0} \frac{(\sqrt{a + x} - \sqrt{1 - x})}{2 x} \times \frac{(\sqrt{1 + x} - \sqrt{1 - x})}{(\sqrt{1 + x} + \sqrt{1 - x})}$

$= {lim}_{x \to 0} \frac{(1 + x) - (1 - x)}{(\sqrt{1 + x} + \sqrt{1 - x})}$

$= {lim}_{x \to 0} \frac{2 x}{2 x (\sqrt{1 + x} + \sqrt{1 - x})}$

$= {lim}_{x \to 0} \frac{1}{(\sqrt{1 + x} + \sqrt{1 - x})}$

$= \frac{1}{\sqrt{1} + \sqrt{1}}$

$= \frac{1}{2}$

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Similar questions

lim x \to 0 \frac{\sqrt{1 + x} - \sqrt{1 - x}}{2 x}

equals

Q. Arrange the following limits in the ascending order
a)

lim x \to 0 {(\frac{1^{x} + 2^{x} + 4^{x}}{3})}^{\frac{1}{x}}

lim x \to 0 {(\frac{2^{x} + 3^{x} + 8^{x} + 27^{x}}{4})}^{\frac{1}{x}}

lim x \to 0 {⎡ ⎣ \frac{1^{x} + (\frac{1}{2})^{x} + (\frac{1}{4})^{x}}{3} ⎤ ⎦}^{\frac{1}{x}}

lim x \to 0 {⎡ ⎣ \frac{(\frac{1}{2})^{x} + (\frac{1}{3})^{x} + (\frac{1}{8})^{x} + (\frac{1}{27})^{x}}{4} ⎤ ⎦}^{\frac{1}{x}}

Evaluate

{lim}_{x \to 0} [\frac{x + 1}{x + 2}]^{2 x + 1}

Evaluate the following one sided limits:

$(i) {lim}_{x \to 2^{+}} \frac{x - 3}{x^{2} - 4}$

$(i i) {lim}_{x \to 2^{-}} \frac{x - 3}{x^{2} - 4}$

$(i i i) {lim}_{x \to 0^{+}} \frac{1}{3 x}$

$(i v) {lim}_{x \to 8^{+}} \frac{2 x}{x + 8}$

$(v) {lim}_{x \to 0^{+}} \frac{2}{x^{\frac{1}{5}}}$

$(v i) {lim}_{x \to \frac{π^{-}}{2}} t a n x$

$(v i i) {lim}_{x \to \frac{π}{2^{+}}} s e c x$

$(v i i i) {lim}_{x \to 0^{-}} \frac{x^{2} - 3 x + 2}{x^{3} - 2 x^{2}}$

$(i x) {lim}_{x \to - 2^{+}} \frac{x^{2} - 1}{2 x + 4}$

$(x) {lim}_{x \to 0^{+}} (2 - c o t x)$

$(x i) {lim}_{x \to 0^{-}} 1 + c o s e c x$

Q. Find

lim x \to 0 f (x)

and

lim x \to 1 f (x)

where

f (x) =

{\begin{matrix} 2 x + 3, x \leq 0 3 (x + 1), x > 0 \end{matrix}

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limx→0√1+x−√1−x2x

limx→0√1+x−√1−x2x Rationalising the numerator =limx→0(√a+x−√1−x)2x×(√1+x−√1−x)(√1+x+√1−x) =limx→0(1+x)−(1−x)(√1+x+√1−x) =limx→02x2x(√1+x+√1−x) =limx→01(√1+x+√1−x) =1√1+√1 =12

${lim}_{x \to 0} \frac{\sqrt{1 + x} - \sqrt{1 - x}}{2 x}$