Problem 132 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(f(a)=g(a)=k\) and their \(n^{\text {th }}\) derivatives \(f^{n}(a)\), \(g^{n}(a)\) exist and are not equal for some \(n .\) Further if \(\lim _{x \rightarrow a} \frac{f(a) g(x)-f(a)-g(a) f(x)+g(a)}{g(x)-f(x)}=4\), then the value of \(k\) is (A) 4 (B) 2 (C) 1 (D) 0

Step-by-Step Solution

Answer

The value of k is 4.

Step 1: Analyze the Given Limits

We start with the given expression \( \lim _{x \rightarrow a} \frac{f(a) g(x)-f(a)-g(a) f(x)+g(a)}{g(x)-f(x)}=4 \). This expression involves limits, so our goal is to simplify it to find \( k \).

Step 2: Simplify the Expression

Rewrite the numerator: \( f(a)g(x) - f(a) - g(a)f(x) + g(a) \) as \( f(a)(g(x) - 1) - g(a)(f(x) - 1) \). This suggests that we see how each function approaches as \( x \rightarrow a \).

Step 3: Application of Limits for Functions

We assume \( f(x) = k + a_n(x-a)^n + \cdots \) and \( g(x) = k + b_n(x-a)^n + \cdots \), where the higher order terms exist because derivatives are not equal at \( n \). For small \( x-a \), these power series can be used.

Step 4: Substitute Approximation in Limit

Use the approximations and substitute in the limit: \[ \frac{k(b_n(x-a)^n) - k - k(a_n(x-a)^n) + k}{k + b_n(x-a)^n - (k + a_n(x-a)^n)}.\] The numerator simplifies to \( k(b_n - a_n)(x-a)^n \), and the denominator simplifies to \( (b_n(x-a)^n - a_n(x-a)^n) \).

Step 5: Cancel and Evaluate the Limit

The expression simplifies to \( \frac{k(b_n - a_n)(x-a)^n}{(b_n - a_n)(x-a)^n} \), which further reduces to \( k \) as the \((x-a)^n\) terms cancel out. Thus, the limit evaluates to \( k = 4 \).

Step 6: Compare with Given Limit

The given limit equals 4, which matches our reduced expression. Therefore, the value of \( k \) that satisfies the condition \( \lim _{x \rightarrow a} = 4 \) is \( k = 4 \).

Key Concepts

Limits

In the realm of Calculus, limits are foundational. They help us understand how functions behave as they approach specific points. The expression mentioned, \( \lim _{x \rightarrow a} \), signifies what happens to a function as \( x \) gets arbitrarily close to \( a \). When dealing with complex functions, limits enable us to predict their behavior even when direct substitution is impossible due to undefined values or discontinuities. In the problem provided, the limit explores the expression's behavior as \( x \) approaches \( a \), allowing us to determine the unknown value \( k \) when specific operations are performed on the functions \( f(x) \) and \( g(x) \). Understanding limits assists in simplifying expressions and evaluating function continuity and changes at points where values may not be directly clear.

Derivatives

Derivatives give us insights into how a function changes at any point. They are the rates at which function values change and are fundamental in Calculus for describing changes. For functions \( f \) and \( g \), derivatives like \( f^n(a) \) and \( g^n(a) \) offer a meticulous look into how they behave near \( a \). Calculating derivatives helps identify the behavior of functions over small steps and contribute to forming power series by evaluating these small changes or rates. In this exercise, the derivatives help show how \( f(x) \) and \( g(x) \) vary in tandem and what their combined behavior suggests about the value \( k \) when the limit is applied. This helps in isolating terms and understanding impacts on the overall function evaluation near points of interest.

Power Series

Power series are expressions that expand functions into an infinite sum of terms, typically involving powers of \( (x-a) \). They are crucial for approximating and analyzing functions, particularly when exact value computations are complex. In our context, expressions like \( f(x)=k+a_n(x-a)^n+\cdots \) reflect how functions can be represented as power series around point \( a \). This representation helps in understanding and simplifying the limit process by breaking down complex functions systematically. Power series enable breaking down expressions into simpler terms that are easier to manipulate, especially when using limits or derivatives. They assist in evaluating changes and capturing local behavior through predominant terms, thus illuminating the main behavior of functions near specific points, even in complex limit configurations.