Lim x - 1-5 x-4-x-x-1

Lim_x → 1√(5x 4) √(x)/x 1 Rationalising the numenator =lim_x → 1(√(5x 4) √(x))/x 1×(√(5x 4)+√(x))/(√(5x 4)+√(x)) =lim_x → 1((5x 4) x)/(x 1)(√(5x 4)+√(x)) =4 lim ...

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Byju's Answer

Standard XII

Mathematics

Rationalization Method to Remove Indeterminate Form

lim x → 1√5 x...

Question

${lim}_{x \to 1} \frac{\sqrt{5 x - 4} - \sqrt{x}}{x - 1}$

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Solution

${lim}_{x \to 1} \frac{\sqrt{5 x - 4} - \sqrt{x}}{x - 1}$

Rationalising the numenator

$= {lim}_{x \to 1} \frac{(\sqrt{5 x - 4} - \sqrt{x})}{x - 1} \times \frac{(\sqrt{5 x - 4} + \sqrt{x})}{(\sqrt{5 x - 4} + \sqrt{x})}$

$= {lim}_{x \to 1} \frac{((5 x - 4) - x)}{(x - 1) (\sqrt{5 x - 4} + \sqrt{x})}$

$= 4 {lim}_{x \to 1} \frac{(x - 1)}{(x - 1) (\sqrt{5 x - 4} + \sqrt{x})}$

$= 4 {lim}_{x \to 1} \frac{1}{\sqrt{5 x - 4} + \sqrt{x}}$

$= 4 \times \frac{1}{\sqrt{5 - 4} + \sqrt{1}}$

$= 4 \times \frac{1}{\sqrt{1} + \sqrt{1}}$

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

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Similar questions

${lim}_{x \to 1} \frac{\sqrt{5 x - 4} - - \sqrt{x}}{x^{3} - 1}$

${lim}_{x \to 1} \frac{\sqrt{5 x - 4} - \sqrt{x}}{x^{2} - 1}$

{lim}_{x \to 1} \frac{\sqrt{5 x - 4} - \sqrt{x}}{x^{3} - 1}

Q. Evaluate :

lim x \to 1 \frac{\sqrt{5 x - 4} - \sqrt{x}}{x^{3} - 1}

Q. Find the limits :

i .

lim x \to 1 [\frac{x^{2} + 1}{x + 100}]

i i .

lim x \to 2 [\frac{x^{3} - 4 x^{2} + 4 x}{x^{2} - 4}]

i i i .

lim x \to 2 [\frac{x^{2} - 4}{x^{3} - 4 x^{2} + 4 x}]

i v .

lim x \to 2 [\frac{x^{3} - 2 x^{2}}{x^{2} - 5 x + 6}]

v .

lim x \to 1 [\frac{x - 2}{x^{2} - x} - \frac{1}{x^{3} - 3 x^{2} + 2 x}]

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limx→1√5x−4−√xx−1

limx→1√5x−4−√xx−1 Rationalising the numenator =limx→1(√5x−4−√x)x−1×(√5x−4+√x)(√5x−4+√x) =limx→1((5x−4)−x)(x−1)(√5x−4+√x) =4limx→1(x−1)(x−1)(√5x−4+√x) =4limx→11√5x−4+√x =4×1√5−4+√1 =4×1√1+√1 =42=2

${lim}_{x \to 1} \frac{\sqrt{5 x - 4} - \sqrt{x}}{x - 1}$