Find the limit. qquad lim_(x rarr oo)(ln x)^(2//x) A) 1 B) 0 C) 2 D) e^(2)

Recognize and Simplify Exponent: Recognize the form of the limit and simplify the exponent. lim_(x rarr oo)(ln x)^(2//x) can be rewritten using the property of exponents as e^(ln((ln x)^(2/x))).Further Simplify Exponent: Simplify the exponent further. ln((ln x)^(2/x)) = (2/x) * ln(ln x).Behavior of ln(ln x): Analyze the behavior of ln(ln x) as x approaches infinity. ln(ln x) increases as x increases, but at a slower rate than ln x.Limit Analysis: Consider the limit of (2/x) * ln(ln x) as x approaches infinity. Since ln(ln x) grows slower than ln x and 2/x approaches 0, the product (2/x) * ln(ln x) approaches 0.Apply Exponential Limit: Apply the limit to the exponential function. lim_(x rarr oo) e^((2/x) * ln(ln x)) = e^0 = 1.

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Find the limit.
qquad

lim_(x rarr oo)(ln x)^(2//x)
A) 1
B) 0
C) 2
D)
e^(2)

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

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Algebra 1

Simplify radical expressions involving fractions

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Q. Find the limit.

\newline

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

\newline

1

\newline

0

\newline

2

\newline

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. $\newline$ $\lim_{x \to \infty} e^{\left(\frac{2}{x} \cdot \ln(\ln x)\right)} = e^0 = 1$ .