lim #160; #160;_ x→∞ #160; ( x^ 2 +5x+3 x^ 2 +x+2 #160; ) ^ x lim #160; #160;_ x→∞ #160; e^ x ( x^ 2 +5x+3 x^ 2 x 2 x^ 2 +x+2 ...

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Algebra of Limits

limx→∞x2+5x+3...

Question

$lim x \to \infty {(\frac{x^{2} + 5 x + 3}{x^{2} + x + 2})}^{x}$ .

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Solution

$\begin{matrix} {lim}_{x \to \infty} {(\frac{x^{2} + 5 x + 3}{x^{2} + x + 2})}^{x} {lim}_{x \to \infty} e^{x} (\frac{x^{2} + 5 x + 3 - x^{2} - x - 2}{x^{2} + x + 2}) [∵ {lim}_{x \to a} {[f (x)]}^{g (x)} = {lim}_{x \to a} e^{g (x)} [f (x) - 1]] {lim}_{x \to \infty} e^{x} (\frac{4 x + 1}{x^{2} + x + 2}) T a k i n g x^{2} c o m m o n f a c t o r f r o m d e n o min a t o r {lim}_{x \to \infty} e^{x} . \frac{4 x + 1}{x^{2} (1 + \frac{1}{x} + \frac{2}{x^{2}})} {lim}_{x \to \infty} e^{x} . \frac{x ⎛ ⎝ 4 + \frac{1}{x} ⎞ ⎠}{x^{2} ⎛ ⎝ 1 + \frac{1}{x} + \frac{2}{x^{2}} ⎞ ⎠} = e (\frac{4 + 0}{1 + 0 + 0}) = e^{4} \end{matrix}$

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