# 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
easy
##### Suggested Book
Mathematical Analysis, Second Edition 520pp/PB 2nd Edition (English, Paperback, T. M. Apostol)

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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