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L’Hospital’s Rule: TIFR GS 2017 Part A Problem 5.

Understand the problem

True or False? Suppose \(f(x) \) is a continuosy differentiable function on \( \Bbb R \) such that \( \lim_{x \to \infty} f(x)= 1 \) and \( \lim_{x \to \infty} f'(x)=b\). Then \(b=1 \).
Source of the problem
TIFR GS 2017 Entrance Examination Paper
Topic
L’Hospital’s principle
Difficulty Level
easy
Suggested Book
Mathematical Analysis, Second Edition 520pp/PB 2nd Edition (English, Paperback, T. M. Apostol)

Start with hints

Do you really need a hint? Try it first!

Can you calculate the value of \( \lim_{x \to \infty} f'(x)\) using the value of \( \lim_{x \to \infty} f(x)\).

See you can write \( f(x) \) as \( \frac{e^x f(x)}{e^x} \). Then taking limit on both sides as \(x \to \infty\), we get
\(lim_{x \to \infty} f(x) = lim_{x \to \infty} \frac{e^x f(x)}{e^x} = lim_{x \to \infty} \frac{e^x f(x)+ e^x f'(x)}{e^x} \), using L’Hospital’s rule.
Put the values of \( lim_{x \to \infty}f(x)=1\) in the above equation. What do you get?

You will get that \( \lim_{x \to \infty}f'(x)=0\). Hence the value of \(b=0\). Therefore, the statement is false.

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