Evaluate the limit-limx -3x2 - 14x2 - 1x31-x

Given: lim_x →∞{3x^2 + 14x^2 1}^x^31+x = lim_x →∞{3 + 1x^24 1x^2}^x^3x (1x+1) = lim_x →∞{3 + 1x^24 1x^2}^x^2(1x+1) = ( 34)^∞ = 0 (∵If a^n = 0 where ...

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Standard XII

Mathematics

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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Evaluate the limit: limx→∞{3x2+14x2−1}x31+x

Given: limx→∞{3x2+14x2−1}x31+x =limx→∞⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩3+1x24−1x2⎫⎪ ⎪ ⎪⎬⎪ ⎪ ⎪⎭x3x(1x+1) =limx→∞⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩3+1x24−1x2⎫⎪ ⎪ ⎪⎬⎪ ⎪ ⎪⎭x2(1x+1) =(34)∞=0 (∵If an=0 where 0<a<1)

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