Evaluate the limit-limx-x2-cx-x

We have, lim_x→∞√(x^2+cx) x On rationalising numerator, we get =lim_x→∞(√(x^2+cx) x)(√(x^2+cx)+x)(√(x^2+cx)+x) =lim_x→∞x^2+cx x^2(√(x^2+cx)+x) =lim_x→∞cx(√( ...

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Standard XII

Mathematics

Algebra of Limits

Evaluate the ...

Question

Evaluate the limit:
$lim x \to \infty \sqrt{x^{2} + c x} - x$

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Solution

We have,
$lim x \to \infty \sqrt{x^{2} + c x} - x$

On rationalising numerator, we get

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

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

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

$= lim x \to \infty \frac{c x}{(x \sqrt{1 + \frac{c}{x}} + x)}$

$= lim x \to \infty \frac{c}{(\sqrt{1 + \frac{c}{x}} + 1)}$

When $x \to \infty$ , then $\frac{1}{x} \to 0$

$= lim x \to \infty \frac{c}{(\sqrt{1 + c (0)} + 1)}$

$= \frac{c}{2}$

Therefore,
$lim x \to \infty \sqrt{x^{2} + c x} - x = \frac{c}{2}$

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Algebra of Limits

Standard XII Mathematics

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Evaluate the limit: limx→∞√x2+cx−x

Evaluate the limit:
$lim x \to \infty \sqrt{x^{2} + c x} - x$