Problem 28 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(f: R \rightarrow[0, \infty)\) be such that \(\lim _{x \rightarrow 5} f(x)\) exists and \(\lim _{x \rightarrow 5} \frac{(f(x))^{2}-9}{\sqrt{|x-5|}}=0\) Then \(\lim _{x \rightarrow 5} f(x)\) equals: (a) 0 (b) 1 (c) 2 (d) 3

Step-by-Step Solution

Answer

The limit \(\lim_{x \rightarrow 5} f(x)\) equals 3.

Step 1: Understand the Given Limit

We are given that \( \lim_{x \rightarrow 5} \frac{(f(x))^{2}-9}{\sqrt{|x-5|}} = 0 \). This indicates that as \( x \) approaches 5, the expression \((f(x))^2 - 9\) must approach 0 faster than \(\sqrt{|x-5|}\).

Step 2: Interpret the Expression

The expression \( (f(x))^2 - 9 \) can be rewritten as \( (f(x)-3)(f(x)+3) \). For the limit of the whole expression to be 0, \( f(x)-3 \) must approach 0, meaning \( f(x) \) must approach 3 as \( x \rightarrow 5 \).

Step 3: Conclusion from Expression Behavior

Since \((f(x))^2 - 9\) approaches 0 as \(x\) approaches 5, it implies \((f(x) - 3)(f(x) + 3)\) approaches 0. This means that either \(f(x) - 3\) or \(f(x) + 3\) approaches 0, and since \(f(x)\) is approaching a positive value, \(f(x) - 3 = 0\).

Step 4: Calculate the Limit

From \(f(x) - 3 = 0\), we conclude that \(f(x)\) approaches 3 as \(x \rightarrow 5\). Thus, \(\lim_{x \rightarrow 5} f(x) = 3\).

Step 5: Verify Answer

Based on our calculations, the answer is consistent with option (d), which is \(\lim_{x \rightarrow 5} f(x) = 3\).

Key Concepts

L'Hopital's Rule

L'Hopital's Rule is a valuable tool in calculus used to solve indeterminate limits. This rule can be applied when you're dealing with limits that result in forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). When you encounter these specific cases, L'Hopital's Rule allows you to find the limit by differentiating the numerator and the denominator separately and then taking the limit again.

• Consider a limit \( \lim_{x \to c} \frac{f(x)}{g(x)} \).
• If the initial substitution results in \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), the rule can be applied.
• Differentiation: Find the derivatives of \( f(x) \) and \( g(x) \), denoted as \( f'(x) \) and \( g'(x) \).
• Re-evaluate the limit \( \lim_{x \to c} \frac{f'(x)}{g'(x)} \).

This process can be repeated if necessary until a determinable form is reached. In our initial exercise, L'Hopital's Rule is not directly used but knowing when and how it applies can clarify complex limit scenarios.

Continuity

Continuity is a fundamental concept in calculus that describes a function's behavior as "unbroken" or "smooth." When evaluating limits, understanding continuity can help predict a function's behavior as it approaches a certain point.
A function \( f(x) \) is continuous at a point \( c \) if the following three conditions are met:

• \( f(c) \) is defined.
• \( \lim_{x \to c} f(x) \) exists.
• \( \lim_{x \to c} f(x) = f(c) \).

In the context of the original exercise, while calculating \( \lim_{x \to 5} f(x) \), continuity ensures that the behavior of \( f(x) \) around \( x = 5 \) aligns with the predicted outcome \( f(5) = 3 \). That is, the function does not "jump" or have holes at this point, giving us confidence in our limit evaluation.

Function Behavior

Understanding function behavior is about comprehending how a function responds as its inputs change. It's essential for evaluating limits and predicting function values around certain points. In mathematical analysis, this involves:

• Analyzing expressions such as \( (f(x))^2 - 9 \) to understand output changes as \( x \) nears a certain value.
• Recognizing patterns to deduce what \( f(x) \) approaches.

In our exercise, the function's behavior near \( x = 5 \) was critical. We transformed \( (f(x))^2 - 9 \) into \( (f(x) - 3)(f(x) + 3) \) to decipher that \( f(x) \) must be approaching 3. This understanding of function behavior allowed us to confidently solve for \( \lim_{x \to 5} f(x) = 3 \). Recognizing these patterns and expressions is a skill that helps reveal how functions behave, assisting in limit evaluations like the one discussed. In essence, grasping how small changes in \( x \) affect \( f(x) \) is pivotal in calculus.