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Single Point Continuity

Evaluate the ...

Question

Evaluate the limit:
$lim x \to \infty {\frac{3 x^{2} + 1}{4 x^{2} - 1}}^{\frac{x^{3}}{1 + x}}$

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Solution

Given:
$lim x \to \infty {\frac{3 x^{2} + 1}{4 x^{2} - 1}}^{\frac{x^{3}}{1 + x}}$

$= lim x \to \infty {⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ \frac{3 + \frac{1}{x^{2}}}{4 - \frac{1}{x^{2}}} ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭}^{\frac{x^{3}}{x (\frac{1}{x} + 1)}}$

$= lim x \to \infty {⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ \frac{3 + \frac{1}{x^{2}}}{4 - \frac{1}{x^{2}}} ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭}^{\frac{x^{2}}{(\frac{1}{x} + 1)}}$

$= {(\frac{3}{4})}^{\infty} = 0$

$(∵ If a^{n} = 0 where 0 < a < 1)$

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Single Point Continuity

Standard XII Mathematics

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