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Step-by-step Solution

Find the limit of $x^{2\log \left(x\right)}$ as $x$ approaches $∞$

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Answer

$\frac{∞^{\log \left(∞^2\right)}}{∞^3\log \left(∞\right)^3}$

Step-by-step explanation

Problem to solve:

$\lim_{x\to∞}\left(\frac{x^{2\log\left(x\right)}}{\left(x\cdot \log\left(x\right)\right)^3}\right)$
1

The power of a product is equal to the product of it's factors raised to the same power

$\lim_{x\to∞}\left(\frac{x^{2\log \left(x\right)}}{x^3\log \left(x\right)^3}\right)$
2

Apply the formula: $a\log_{b}\left(x\right)$$=\log_{b}\left(x^a\right)$, where $a=2$ and $b=10$

$\lim_{x\to∞}\left(\frac{x^{\log \left(x^2\right)}}{x^3\log \left(x\right)^3}\right)$

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Answer

$\frac{∞^{\log \left(∞^2\right)}}{∞^3\log \left(∞\right)^3}$
$\lim_{x\to∞}\left(\frac{x^{2\log\left(x\right)}}{\left(x\cdot \log\left(x\right)\right)^3}\right)$

Main topic:

Limits by direct substitution

Time to solve it:

~ 0.02 seconds