ISI MStat 2018 PSA Problem 8 | Limit of a Function

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

Answer: is (C)

ISI MStat 2018 PSA Problem 8

Introduction to Real Analysis by Bertle Sherbert

Try with Hints

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

$\infty^0$
$\frac{\infty}{\infty}$
L'Hospital Rule

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 .

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 .

Similar Problems and Solutions

ISI MStat 2018 PSA Problem 8 — Outstanding Statistics Program with Applications

Outstanding Statistics Program with Applications

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

Answer: is (C)

ISI MStat 2018 PSA Problem 8

Introduction to Real Analysis by Bertle Sherbert

Try with Hints

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

$\infty^0$
$\frac{\infty}{\infty}$
L'Hospital Rule

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 .

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 .

Similar Problems and Solutions

ISI MStat 2018 PSA Problem 8 — Outstanding Statistics Program with Applications

Outstanding Statistics Program with Applications

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