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Limits — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsLimits4 Marks
Question
Evaluate the following limits:
$\lim _{x \rightarrow \infty}\left[\frac{\left(3 x^2+4\right)\left(4 x^2-6\right)\left(5 x^2+2\right)}{4 x^6+2 x^4-1}\right]$
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Answer

$ \lim _{x \rightarrow \infty} \frac{\left(3 x^2+4\right)\left(4 x^2-6\right)\left(5 x^2+2\right)}{4 x^6+2 x^4-1}$
$=\lim _{x \rightarrow \infty} \frac{\frac{\left(3 x^2+4\right)\left(4 x^2-6\right)\left(5 x^2+2\right)}{x^6}}{\frac{4 x^6+2 x^4-1}{x^6}} $
$ =\lim _{x \rightarrow \infty} \frac{\left(\frac{3 x^2+4}{x^2}\right)\left(\frac{4 x^2-6}{x^2}\right)\left(\frac{5 x^2+2}{x^2}\right)}{4+\frac{2}{x^2}-\frac{1}{x^6}}$
$=\frac{\lim _{x \rightarrow \infty}\left(3+\frac{4}{x^2}\right)\left(4-\frac{6}{x^2}\right)\left(5+\frac{2}{x^2}\right)}{\lim _{x \rightarrow \infty}\left(4+\frac{2}{x^2}-\frac{1}{x^6}\right)}$
$=\frac{\lim _{x \rightarrow \infty}\left(3+\frac{4}{x^2}\right) \cdot \lim _{x \rightarrow \infty}\left(4-\frac{6}{x^2}\right) \cdot \lim _{x \rightarrow \infty}\left(5+\frac{2}{x^2}\right)}{\lim _{x \rightarrow \infty} 4+\lim _{x \rightarrow \infty} \frac{2}{x^2}-\lim _{x \rightarrow \infty} \frac{1}{x^6}}$
$=\frac{(3+0)(4-0)(5+0)}{4+0-0} \ldots\left[\lim _{x \rightarrow \infty} \frac{1}{x^{ k }}=0, k >0\right]$
$=\frac{3 \times 4 \times 5}{4}$
$=15 $

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