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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.

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

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

Limit

L'hospital Rule

But try the problem first...

Answer: is (C)

Source

Suggested Reading

ISI MStat 2018 PSA Problem 8

Introduction to Real Analysis by Bertle Sherbert

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 .

What to do to shape your Career in Mathematics after 12th?

From the video below, let's learn from Dr. Ashani Dasgupta (a Ph.D. in Mathematics from the University of Milwaukee-Wisconsin and Founder-Faculty of Cheenta) how you can shape your career in Mathematics and pursue it after 12th in India and Abroad. These are some of the key questions that we are discussing here:

- What are some of the best colleges for Mathematics that you can aim to apply for after high school?
- How can you strategically opt for less known colleges and prepare yourself for the best universities in India or Abroad for your Masters or Ph.D. Programs?
- What are the best universities for MS, MMath, and Ph.D. Programs in India?
- What topics in Mathematics are really needed to crack some great Masters or Ph.D. level entrances?
- How can you pursue a Ph.D. in Mathematics outside India?
- What are the 5 ways Cheenta can help you to pursue Higher Mathematics in India and abroad?

Cheenta has taken an initiative of helping College and High School Passout Students with its "Open Seminars" and "Open for all Math Camps". These events are extremely useful for students who are really passionate for Mathematic and want to pursue their career in it.

Cheenta is a knowledge partner of Aditya Birla Education Academy

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.

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