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Indeterminate Forms

If L=limx→-∞√...

Question

If $L = lim x \to - \infty \sqrt{25 x^{2} - 3 x} + 5 x$ , then the value of $[\frac{1}{L}]$ is
(where $[.]$ denotes greatest integer function)

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Solution

$L = lim x \to - \infty \sqrt{25 x^{2} - 3 x} + 5 x, (\infty - \infty form) = lim x \to - \infty \frac{25 x^{2} - 3 x - 25 x^{2}}{\sqrt{25 x^{2} - 3 x} - 5 x} = lim x \to - \infty \frac{- 3 x}{- x (\sqrt{25 - \frac{3}{x}}) - 5 x} (∵ x \to - \infty \Rightarrow | x | = - x) = \frac{3}{\sqrt{25 - 0} + 5} = \frac{3}{10} ∴ [\frac{1}{L}] = 3$

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Indeterminate Forms

Standard XII Mathematics

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