What is the limit as x approaches infinity of ((2ln(x))/(ln(7x^2)))^{ln(x)} ?

The limit as x approaches infinity of ((2ln(x))/(ln(7x^2)))^{ln(x)} is 1/(sqrt(7))

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limit as x approaches infinity of ((2ln(x))/(ln(7x^2)))^{ln(x)}

x \to \infty lim ((\frac{2 ln ( x )}{ln ( 7 x ^{2} )})^{l n (x)})

\frac{1}{7}

Decimal

0.37796 \dots

Solution steps

x \to \infty lim ((\frac{2 ln ( x )}{ln ( 7 x ^{2} )})^{l n (x)})

Apply exponent rules

= x \to \infty lim (e^{l n (x) l n (\frac{2 l n ( x )}{l n ( 7 x ^{2} )})})

Apply the Limit Chain Rule:

\frac{1}{7}

= \frac{1}{7}

x \to - \frac{π}{2} lim (- 6 x - \frac{π}{2})

x \to - 6 lim (\frac{4}{x + 6})

x \to 2 lim (x^{2} - 2 x + 2)

x \to \frac{π}{6} lim (2^{s i n (x)})

x \to π - lim (tan (x))

What is the limit as x approaches infinity of ((2ln(x))/(ln(7x^2)))^{ln(x)} ?
The limit as x approaches infinity of ((2ln(x))/(ln(7x^2)))^{ln(x)} is 1/(sqrt(7))