Evaluate the following limits : _ x [7 x^2+2 x-3 x^4+x+2 ]

Question

Evaluate the following limits : $\lim _{x \rightarrow \infty}\left[\frac{7 x^2+2 x-3}{\sqrt{x^4+x+2}}\right]$

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Answer

$\lim _{x \rightarrow \infty}\left[\frac{7 x^2+2 x-3}{\sqrt{x^4+x+2}}\right]$
$=\lim _{x \rightarrow \infty} \frac{\frac{7 x^2+2 x-3}{x^2}}{\frac{\sqrt{x^4+x+2}}{x^2}} \cdots[$Divide numerator and denominator by $x^2]$
$=\frac{\lim _{x \rightarrow \infty}\left(7+\frac{2}{x}-\frac{3}{x^2}\right)}{\lim _{x \rightarrow \infty} \sqrt{\frac{x^4+x+2}{x^4}}}$
$=\frac{\lim _{x \rightarrow \infty} 7+\lim _{x \rightarrow \infty} \frac{2}{x}-\lim _{x \rightarrow \infty} \frac{3}{x^2}}{\lim _{x \rightarrow \infty} \sqrt{1+\frac{1}{x^3}+\frac{2}{x^4}}} \ldots\left[\lim _{x \rightarrow \infty} \frac{1}{x^2}=0, \mathrm{k}>0\right]$
$=\frac{7+0-0}{\sqrt{1+0+0}} 7$
$=7$

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