Problem 118 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

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

Step-by-Step Solution

Answer

The value of \( k \) is 4.

Step 1: Setup the Limit Expression

First, identify the expression given in the limit. We have:\[ \lim_{x \rightarrow a} \frac{f(a)g(x) - f(a) - g(a)f(x) + f(a)}{g(x) - f(x)} = 4 \]Substitute \( f(a) = g(a) = k \) to simplify the expression.

Step 2: Simplify Using Known Values

Substitute \( f(a) = g(a) = k \) into the limit expression:\[ \lim_{x \rightarrow a} \frac{k g(x) - k - k f(x) + k}{g(x) - f(x)} \]The expression simplifies to:\[ \lim_{x \rightarrow a} \frac{k(g(x) - f(x))}{g(x) - f(x)} \]

Step 3: Further Simplify the Expression

Continue to simplify:\[ \lim_{x \rightarrow a} \frac{k(g(x) - f(x))}{g(x) - f(x)} = \lim_{x \rightarrow a} k \]This indicates \( k \) if it is well-defined as long as \( g(x) eq f(x) \).

Step 4: Solve for the Value of k

Since the limit equals 4, we have:\[ k = 4 \]Therefore, the value of \( k \) that satisfies the given limit is 4.

Key Concepts

nth derivatives

The concept of nth derivatives is a cornerstone in calculus that involves taking derivatives multiple times. If you’ve ever taken the derivative of a function, you’ve found what is called the first derivative. This shows the rate of change of a function at any point. To find the nth derivative, you keep differentiating the function repeatedly.

• The zero-th derivative is simply the function itself.
• The first derivative is the slope of the function.
• The n-th derivative is obtained by taking the derivative of the (n-1)-th derivative.

Understanding nth derivatives helps in analyzing the behavior of functions and solving complex calculus problems. For instance, if the function and its n-th derivatives exist and differ for some value, it indicates certain differences in the function's growth or shrinkage near a point. This concept is crucial in the interplay with limits and differentiability discussed in the exercise.

limit expression

A limit expression tries to evaluate the behavior of a function as the input approaches a certain point. In this context, the limit expression is set for when \( x \) approaches \( a \). Understanding limits is crucial because many real-world phenomena can be described using limits.When evaluating limit expressions:

• Identify what value the function approaches as \( x \) approaches a particular point.
• If you encounter a direct substitution leading to an indeterminate form, additional simplification or techniques like L'Hopital's Rule may be needed.

In our problem, the limit \( \lim_{x \rightarrow a} \frac{f(a)g(x) - f(a) - g(a)f(x) + f(a)}{g(x) - f(x)} \) simplifies using known values, revealing valuable insights about the function's behavior at that point, specifically helping determine the value of \( k \).

simplifying expressions

Simplifying mathematical expressions is like cleaning up an equation to make it easier to understand and solve. It involves cancelling terms, factoring, and condensing complex expressions into their simplest form.In our problem:

• Substitute known values like \( f(a) = g(a) = k \) to reduce complexity.
• Simplify expressions by factoring or cancelling common terms.

The primary goal is to make the mathematical operations straightforward and accessible. When you simplify the given expression step by step, it guides you towards the solution. This helps in resolving the complex parts and understanding the underlying function relationships more easily. Understanding how to simplify effectively is a critical skill for solving calculus problems.

calculus problems

Solving calculus problems often involves several steps and uses different concepts together. Calculus helps to explore changes in mathematical properties and is widely applicable in physics, engineering, economics, and more. Key approaches when tackling calculus problems include:

• Understanding the problem and what it requires—identify what is being asked.
• Breaking down a complex problem into simpler parts, like leveraging known derivatives or limits.
• Using clear, logical steps to reach a solution—like setting up limit expressions and solving using stepwise simplifications.

Our specific exercise involved derivatives, limits, and simplification. By organizing these steps logically and ensuring you understand each concept involved, solutions to calculus problems become more manageable and clearer. Mastery in this field of mathematics opens doors to advanced mathematical applications and problem-solving skills.