Problem 32 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(f\) is differentiable function satisfying \(f(0)=0\), and if \(g(x)=\frac{f(x)}{x}\), then the value, that should be assigned to \(g(0)\), so that \(g\) is continuous at ' 0 ' is (A) \(]\) (B) 0 (C) \(f(0)\) (D) \(f^{\prime}(0)\)

Step-by-Step Solution

Answer

Assign \( g(0) = f'(0) \), choice (D).

Step 1: Understand the Problems and Conditions

We need to find the value to assign to \( g(0) \) so that the function \( g(x) = \frac{f(x)}{x} \) remains continuous at \( x = 0 \). Given, \( f(0) = 0 \) and \( f \) is differentiable.

Step 2: Use the Definition of Continuity

For \( g(x) \) to be continuous at \( x = 0 \), it must satisfy the condition: \( \lim_{{x \to 0}} g(x) = g(0) \). This implies \( \lim_{{x \to 0}} \frac{f(x)}{x} \) should equal \( g(0) \).

Step 3: Apply L'Hopital's Rule on \( \lim_{{x \to 0}} \frac{f(x)}{x} \)

Since \( \lim_{{x \to 0}} \frac{f(x)}{x} \) is in the indeterminate form \( \frac{0}{0} \), we apply L'Hopital's Rule: \[ \lim_{{x \to 0}} \frac{f(x)}{x} = \lim_{{x \to 0}} \frac{f'(x)}{1} = f'(0) \].

Step 4: Assign the Correct Value to \( g(0) \) Based on the Limit

From Step 3, \( \lim_{{x \to 0}} \frac{f(x)}{x} = f'(0) \). Thus, for \( g(x) \) to be continuous at \( x = 0 \), assign \( g(0) = f'(0) \). Therefore, the correct choice is (D) \( f'(0) \).

Key Concepts

Continuity

Continuity is a fundamental concept in calculus that describes how functions behave around a point. For a function to be continuous at a given point, the function must be well-defined, and the limit as we approach the point must equal the function's value at that point. In simpler terms, you shouldn't have to "lift your pencil" when drawing the function. This means, mathematically, that if you have a function \( g(x) \) and you want to make sure it is continuous at \( x = 0 \), you need:

• \( g(0) \) to exist
• The limit \( \lim_{{x \to 0}} g(x) \) to exist
• \( \lim_{{x \to 0}} g(x) = g(0) \)

In the exercise, we are given that \( g(x) = \frac{f(x)}{x} \). For \( g(x) \) to be continuous at \( x = 0 \), we needed the limit \( \lim_{{x \to 0}} \frac{f(x)}{x} \) to equal \( g(0) \). That's why continuity is an integral part of solving the given problem.

L'Hopital's Rule

L'Hopital's Rule is a powerful tool in calculus used to find limits of indeterminate forms. An indeterminate form often appears when both the numerator and the denominator of a fraction approach zero or infinity. When you stumble across such limits, L'Hopital's Rule says you can take the derivative of the numerator and denominator separately and then find the limit again.In our exercise, we had the limit \( \lim_{{x \to 0}} \frac{f(x)}{x} \). This expression is of the form \( \frac{0}{0} \), so L'Hopital's Rule can be applied. By differentiating the numerator \( f(x) \) and the denominator \( x \), we get:

• Numerator's derivative: \( f'(x) \)
• Denominator's derivative: 1

Thus, the rule gives us \( \lim_{{x \to 0}} \frac{f(x)}{x} = \lim_{{x \to 0}} \frac{f'(x)}{1} = f'(0) \). This result guided us in determining the value for \( g(0) \) to maintain continuity.

Limits

Limits help us understand the behavior of functions as they approach a particular point. They are indispensable in defining key concepts such as continuity, derivatives, and integrals. A limit tells us what value a function approaches as its input approaches a certain point.For the function \( g(x) = \frac{f(x)}{x} \), finding the limit \( \lim_{{x \to 0}} \frac{f(x)}{x} \) was crucial to determine if \( g(x) \) could be made continuous at \( x = 0 \). Since \( f(0) = 0 \), initially both the numerator and denominator go to zero as \( x \to 0 \), resulting in an indeterminate form \( \frac{0}{0} \). That's where understanding limits and applying L'Hopital's Rule specifically comes in handy to resolve the indeterminacy and find the actual limit.It’s these detailed investigations into limits that helped solve the main problem by confirming \( g(0) \) should equal \( f'(0) \) for continuity to hold at \( x = 0 \). Learning about limits prepares you to tackle a wide array of calculus problems by examining function behavior at critical points.