lim_x→∞(10 e^3x+8)^5/x ⇒lim_x→∞ exp(log ((10e^3x+8)^5/x)) =lim_x→∞ exp[s log(10 e^3x+8)x] ⇒ exp[lim_x→∞(s log(10e^3x+8)x)] =exp[5 lim_x ...

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Algebra of Limits

limx→∞ 10e3x ...

Question

$l i m_{x \to \infty} (10 e^{3 x} + 8)^{5 / x}$

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Solution

$lim x \to \infty (10 e^{3 x} + 8)^{5 / x}$
$\Rightarrow {lim}_{x \to \infty} e x p (l o g ((10 e^{3 x} + 8)^{5 / x}))$
$= lim x \to \infty e x p [\frac{s l o g (10 e^{3 x} + 8)}{x}]$
$\Rightarrow e x p [lim x \to \infty (\frac{s l o g (10 e^{3 x} + 8)}{x})]$
$= e x p ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 5 lim x \to \infty ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ \frac{l o g (10 e^{3 x}) + l o g (\frac{4 e^{- 3 x}}{5} + 1)}{x} ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ ⎤ ⎥ ⎥ ⎥ ⎥ ⎦$
but $lim x \to \infty l o g (\frac{4 e^{- 3 x}}{5} + 1) = 0$
$\Rightarrow e x p [5 lim x \to \infty (\frac{l o g (10 e^{3 x})}{x})]$
Applying L' Hospital's rule
$e x p ⎡ ⎢ ⎢ ⎢ ⎣ 5 lim x \to 0 \frac{d}{d x} \frac{l o g (10 e^{3 x})}{\frac{d x}{d x}} ⎤ ⎥ ⎥ ⎥ ⎦$
$= e x p (5 lim x \to \infty \frac{3}{1}) = e x p (5 \times 3) = e^{15}$ .

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