Lim_x →∞√(x^2+7x) x On Rationalising, we get =lim_x →∞ ( (√(x^2+7x) x ) (√(x^2+7x)+x )/ (√(x^2+7x)+x ) ) =lim_x →∞ ( (x^2+7x) x^2/√(x^2+7x)+x ) =lim_x →∞7 ...

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Byju's Answer

Standard XII

Mathematics

Rationalization Method to Remove Indeterminate Form

lim x →∞√x2+7...

Question

${lim}_{x \to \infty} \sqrt{x^{2} + 7 x} - x$

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Solution

${lim}_{x \to \infty} \sqrt{x^{2} + 7 x} - x$

On Rationalising, we get

$= {lim}_{x \to \infty} (\frac{(\sqrt{x^{2} + 7 x} - x) (\sqrt{x^{2} + 7 x} + x)}{(\sqrt{x^{2} + 7 x} + x)})$

$= {lim}_{x \to \infty} (\frac{(x^{2} + 7 x) - x^{2}}{\sqrt{x^{2} + 7 x} + x})$

$= {lim}_{x \to \infty} \frac{7 x}{\sqrt{x^{2} + 7 x} + x}$

$= {lim}_{x \to \infty} \frac{7 x}{\sqrt{x^{2} + (1 + \frac{7 x}{x^{2}})} + x}$

$= {lim}_{x \to \infty} \frac{7 x}{x \sqrt{1 + \frac{7}{x}} + x}$

$a s x \to \infty, \frac{1}{x} \to 0$

$= {lim}_{x \to \infty} \frac{7}{\sqrt{1 + \frac{7}{x}} + 1}$

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limx→∞√x2+7x−x

limx→∞√x2+7x−x On Rationalising, we get =limx→∞((√x2+7x−x)(√x2+7x+x)(√x2+7x+x)) =limx→∞((x2+7x)−x2√x2+7x+x) =limx→∞7x√x2+7x+x =limx→∞7x√x2+(1+7xx2)+x =limx→∞7xx√1+7x+x as x→∞,1x→0 =limx→∞7√1+7x+1

${lim}_{x \to \infty} \sqrt{x^{2} + 7 x} - x$