The function f(x) = x^{frac{1}{ln x}} is: | vedclass

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(D) Given the function $f(x) = x^{\frac{1}{\ln x}}$.For the function to be defined,we must have $x > 0$ and $x 
eq 1$ (since $\ln x$ is in the denomi

Vedclass · Accepted Answer

both $(A)$ and $(C)$

Vedclass · Answer

(D) Given the function $f(x) = x^{\frac{1}{\ln x}}$.For the function to be defined,we must have $x > 0$ and $x 
eq 1$ (since $\ln x$ is in the denominator and $\ln x 
eq 0$).Taking the natural logarithm on both sides:$\ln(f(x)) = \ln(x^{\frac{1}{\ln x}}) = \frac{1}{\ln x} \cdot \ln x = 1$.Thus,$\ln(f(x)) = 1$,which implies $f(x) = e^1 = e$.Since $f(x) = e$ for all $x$ in its domain $(0, 1) \cup (1, \infty)$,it is a constant function.Also,$\lim_{x 	o 1} f(x) = \lim_{x 	o 1} e = e$,so the limit exists.Therefore,both $(A)$ and $(C)$ are correct.

The function $f(x) = x^{\frac{1}{\ln x}}$ is:

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