Find the limit. $\newline$ $\lim_{x \to \infty}(\ln x)^{\frac{2}{x}}$ $\newline$ A) $1$ $\newline$ B) $0$ $\newline$ C) $2$ $\newline$ D) $e^{2}$

Full solution

Q. Find the limit.

\newline

\lim_{x \to \infty}(\ln x)^{\frac{2}{x}}

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1

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0

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2

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e^{2}

Recognize and Simplify Exponent: Recognize the form of the limit and simplify the exponent. $\lim_{x \rightarrow \infty}(\ln x)^{\frac{2}{x}}$ can be rewritten using the property of exponents as $e^{\ln((\ln x)^{\frac{2}{x}})}$ .
Further Simplify Exponent: Simplify the exponent further. $\newline$ $\ln((\ln x)^{\frac{2}{x}}) = \frac{2}{x} \cdot \ln(\ln x)$ .
Behavior of $\ln(\ln x)$ : Analyze the behavior of $\ln(\ln x)$ as $x$ approaches infinity. $\newline$ $\ln(\ln x)$ increases as $x$ increases, but at a slower rate than $\ln x$ .
Limit Analysis: Consider the limit of $(\frac{2}{x}) \cdot \ln(\ln x)$ as $x$ approaches infinity. Since $\ln(\ln x)$ grows slower than $\ln x$ and $\frac{2}{x}$ approaches $0$ , the product $(\frac{2}{x}) \cdot \ln(\ln x)$ approaches $0$ .
Apply Exponential Limit: Apply the limit to the exponential function. $\newline$ $\lim_{x \to \infty} e^{\left(\frac{2}{x} \cdot \ln(\ln x)\right)} = e^0 = 1$ .