Problem 61 - 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 value, that should be assigned to \(g(0)\), so that \(g\) is continuous at ' 0 ' is (A) 1 (B) 0 (C) \(f(0)\) (D) \(f^{\prime}(0)\)

Step-by-Step Solution

Answer

(D) \(f^{\prime}(0)\)

Step 1: Understanding the problem

We need to find the value that should be assigned to \(g(0)\) so that \(g(x)=\frac{f(x)}{x}\) is continuous at \(x = 0\). Given that \(f(0)=0\) and \(f\) is differentiable, we must compute the limit \(\lim_{x \to 0} \frac{f(x)}{x}\).

Step 2: Limit of the difference quotient

Since \(f\) is differentiable at \(0\), we have the derivative \(f^{\prime}(0) = \lim_{x \to 0} \frac{f(x) - f(0)}{x} = \lim_{x \to 0} \frac{f(x)}{x}\) because \(f(0) = 0\).

Step 3: Assigning the limit value to ensure continuity

For \(g(x)\) to be continuous at \(x = 0\), the value of \(g(0)\) must equal \(\lim_{x \to 0} \frac{f(x)}{x}\). From the previous step, this limit is equal to \(f^{\prime}(0)\).

Step 4: Final conclusion and answer choice

The value to assign to \(g(0)\) for continuity is \(f^{\prime}(0)\). Therefore, the correct choice is (D): \(f^{\prime}(0)\).

Key Concepts

Continuity

Continuity is a foundational concept in calculus that helps us understand how functions behave and how "smoothly" they act across their domain. A function \(g(x)\) is said to be continuous at a point \(x = a\) if the following three conditions are met:

• The function \(g(x)\) is defined at \(x = a\).
• The limit of the function \(g(x)\) as \(x\) approaches \(a\) exists.
• The function's value at \(a\) matches the limit, i.e., \(g(a) = \lim_{x \to a} g(x)\).

These conditions ensure there is no "jump" or "hole" at \(x = a\). In the given exercise, the function \(g(x) = \frac{f(x)}{x}\) is defined everywhere except for \(x = 0\), where it becomes undefined due to division by zero. However, we are given that \(f(x)\) is differentiable and that its derivative can help us evaluate continuity at \(0\). By assigning the value of the limit of \(g(x)\) as \(x\) approaches \(0\), which is \(f'(0)\), to \(g(0)\), we make \(g(x)\) continuous at this point, satisfying all conditions of continuity.

Derivative

The derivative is a concept that provides us with a way to measure how a function changes at each point in its domain. If a function \(f(x)\) is differentiable at \(x = 0\), it means that we can compute the derivative, written as \(f'(0)\). This derivative represents the instantaneous rate of change of the function near \(x = 0\). Derivatives can be intuitively thought of as the slope of the tangent line to the curve of the function at any given point. In our exercise, we see that to determine the value of \(g(0)\), we rely on the fact that \(f(x)\) is differentiable at \(0\). Here, \(f'(0)\) plays a crucial role since \( \lim_{x \to 0} \frac{f(x) - f(0)}{x} = f'(0)\). Given that \(f(0) = 0\), this simplifies straight to the limit \( \lim_{x \to 0} \frac{f(x)}{x} = f'(0)\), which becomes the value we assign to \(g(0)\) for continuity purposes.

Limit of a Function

The concept of limits is central to calculus and provides a way to understand behavior at the boundaries of function domains. A limit answers the question: "What value does the function approach as it gets close to a certain point?"For example, in the context of the exercise, we need to determine the limit of \(\frac{f(x)}{x}\) as \(x\) approaches \(0\). Because division by zero is undefined, we instead look at what the function approaches as \(x\) gets very close to zero from both sides. Since \(f\) is differentiable at zero, we are guaranteed that this limit exists and equals \(f'(0)\). This is why differentiability is so useful—it gives us both existence and a way to calculate these otherwise uncertain values. Using the limit \( \lim_{x \to 0} \frac{f(x)}{x} = f^{\prime}(0)\), we can confidently assign this limit as the value of \(g(0)\), ensuring the function \(g(x)\) is continuous there.