This is a beautiful problem from ISI MStat 2018 PSA Problem 8 based on limit of a function. Try yourself and use hints if required.

## Limit of a Function – ISI MStat Year 2018 PSA Question 8

The value of \( \lim _{x \rightarrow \infty}(\log x)^{1 / x} \)

- (A) is e
- (B) is 0
- (C) is 1
- (D) does not exist

**Key Concepts**

Limit

L’hospital Rule

## Check the Answer

But try the problem first…

Answer: is (C)

ISI MStat 2018 PSA Problem 8

Introduction to Real Analysis by Bertle Sherbert

## Try with Hints

First hint

What is the form of the limit? Can you convert it to some known limit?

\(\infty^0\)

\(\frac{\infty}{\infty}\)

L’Hospital Rule

Second Hint

Let , \( \lim _{x \rightarrow \infty}(\log x)^{1 / x} \) =l (say) then taking log on both sides we get , \( \lim _{x \rightarrow \infty} \frac{ log( logx) }{x} \) =log (l) . Now we will apply L’hospital rule .

Final Step

Applying L’hospital rule we get , log (l)= \( \lim _{x \rightarrow \infty} \frac{1}{logx x} \)= 0 \( \Rightarrow l=e^0 \Rightarrow l=1 \)

Hence , option (C) is correct .

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