Question

lim_{x rightarrow infty} x^{frac{1}{ln x}}Select one:a 1b 0c ∞d e

Functions
ANSWERED
asked 2020-12-25

\(\lim_{x \rightarrow \infty} x^{\frac{1}{ln x}} \)Select one:
a 1
b 0
c ∞
d e

 

Answers (1)

2020-12-26

Let LL be the limit. Then:\(\begin{aligned} L&,=\lim_{x\to\infty}x^{1/,\ln x}\\ \ln L&,=\lim_{x\to\infty}\ln x^{1/,\ln x}&,&,\text{Take the natural log of both sides.}\\ \ln L&,=\lim_{x\to\infty}\left(\frac{1}{\ln x}\right)\ln x&,&,\text{Use the rule $\ln x^a=a\ln x$.}\\ \ln L&,=\lim_{x\to\infty}1&,&,\text{Simplify.}\\ \ln L&,=1&,&,\text{Evaluate the limit.}\\ e^{\ln L}&,=e^1&,&,\text{Raise both sides as a power of $e$.}\\ L&,=e&,&,\text{Use the rule $e^{\ln x}=x$.} \end{aligned}\)

The correct answer is then d. e d. e ​ . ​ ​

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