Select Page

# 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.

# Connected Program at Cheenta

#### College Mathematics Program

The higher mathematics program caters to advanced college and university students. It is useful for I.S.I. M.Math Entrance, GRE Math Subject Test, TIFR Ph.D. Entrance, I.I.T. JAM. The program is problem driven. We work with candidates who have a deep love for mathematics. This program is also useful for adults continuing who wish to rediscover the world of mathematics.

# Similar Problems

## Isomorphism in b/w infinite dim vector sp: TIFR GS 2019, Part B Problem 10

It is a question on isomomorphisms b/w inf dim vector spaces. It was asked in TIFR 2019 GS admission paper. It is a true false question.

## Matrix to real line: TIFR GS 2019, Part B Problem 8

It is a lie algebra question on connections b/w matrices and real space. It was asked in TIFR 2019 GS admission paper. It is a true false question.

## Homomorphism to Continuous function: TIFR GS 2019, Part B Problem 9

It is a lie algebra question on homomorphisms b/w real ring and ring of continuous function. It was asked in TIFR 2019 GS admission paper. It is a true false question.

## Similar matrices: TIFR GS 2019, Part B Problem 7

It is a linear algebra question on similar matrices. It was asked in TIFR 2019 GS admission paper. It is a true false question.

## Average Determinant: TIFR GS 2017 Part A Problem 8.

This question has appeared in TIFR GS 2017 Entrance Examination and is based on Linear Algebra.

## Spanning set of a matrix space: TIFR GS 2019, Part B Problem 6

It is a linear algebra question on matrices basically on spanning set of a matrix space. It was asked in TIFR 2019 GS admission paper.

## ABC of rank: TIFR GS 2019, Part B Problem 5

It is a linear algebra question on matrices. It was asked in TIFR 2019 GS admission paper. It is a true false question.

## Continuous map on countable space: TIFR GS 2019, Part B Problem 3

It is a topology question on real plane. It was asked in TIFR 2019 GS admission paper. It is true-false type question.

## Invertible Matrix implies identity?: TIFR GS 2019, Part B Problem 4

It is a linear algebra question on matrices. It was asked in TIFR 2019 GS admission paper.

## Invertible Matrix: TIFR GS 2019, Part B Problem 2

It is a linear algebra question on matrices. It was asked in TIFR 2019 GS admission paper.