Lim_x →∞x/√(4x^2+1) 1 lim_x →∞ [x/√(4x^2+1) 1 ] Rationalising the denominator: lim_x →∞ [ x/ ( √(4x^2+1) 1 ) ( √(4x^2+1)+1 )/ ( √(4x^2+1)+1 ) ] =lim_x →∞ [ x ...

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Rationalization Method to Remove Indeterminate Form

lim x →∞x/√4 ...

Question

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

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Solution

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

${lim}_{x \to \infty} [\frac{x}{\sqrt{4 x^{2} + 1} - 1}]$

Rationalising the denominator:

${lim}_{x \to \infty} [\frac{x}{(\sqrt{4 x^{2} + 1} - 1)} \frac{(\sqrt{4 x^{2} + 1} + 1)}{(\sqrt{4 x^{2} + 1} + 1)}]$

$= {lim}_{x \to \infty} [\frac{x (\sqrt{4 x^{2} + 1} + 1)}{4 x^{2} + 1 - 1}]$

$= {lim}_{x \to \infty} [\frac{\sqrt{4 x^{2} + 1} + 1}{4 x}]$

Dividing the numerator and the denominator by x:

$= {lim}_{x \to \infty} [\frac{\frac{\sqrt{4 x^{2} + 1}}{x} + \frac{1}{x}}{4}]$

$= {lim}_{x \to \infty} ⎡ ⎢ ⎣ \frac{\frac{\sqrt{4 x^{2} + 1}}{x^{2}} + \frac{1}{x}}{4} ⎤ ⎥ ⎦$

$= {lim}_{x \to \infty} ⎡ ⎢ ⎣ \frac{\sqrt{4 + \frac{1}{x^{2}}} + \frac{1}{x}}{4} ⎤ ⎥ ⎦$

$x \to \infty$

$∴ \frac{1}{x}, \frac{1}{x^{2}} \to 0 = \frac{\sqrt{4}}{4} = \frac{2}{4} = \frac{1}{2}$

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limx→∞x√4x2+1−1

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