Problem 3 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The value of \(\lim _{x \rightarrow \infty}(\sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x})\) is (A) \(\frac{1}{2}\) (C) 0 (D) None of these

Step-by-Step Solution

Answer

The limit is \(\frac{3}{4}\), so the answer is (D) None of these.

Step 1: Rewrite the Expression

The problem requires us to find the limit as \(x\) approaches infinity of the expression \(\sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x}\). This expression is difficult in its current form, so we need to simplify it for analysis.

Step 2: Simplify the Inside of the Square Root

First, note that \(\sqrt{x+\sqrt{x}}\) can be approximated as \(\sqrt{x} + \frac{1}{2}\sqrt{x}\) for large \(x\). Then \(x + \sqrt{x+\sqrt{x}}\) becomes \(x + \sqrt{x} + \frac{1}{2}\sqrt{x}\), which can be further approximated by \(x + \frac{3}{2}\sqrt{x}\).

Step 3: Use Binomial Expansion

Now, use the binomial expansion to approximate \(\sqrt{x + \frac{3}{2}\sqrt{x}}\). This gives us \(\sqrt{x}\sqrt{1 + \frac{3}{2}\frac{1}{\sqrt{x}}} \approx \sqrt{x} + \frac{1}{2}\cdot \frac{3}{2}\sqrt{x}\cdot \frac{1}{\sqrt{x}} = \sqrt{x} + \frac{3}{4}\).

Step 4: Calculate the Limit

Substitute the approximation back into the original expression: \(\sqrt{x + \frac{3}{2}\sqrt{x}} - \sqrt{x} = \left(\sqrt{x} + \frac{3}{4}\right) - \sqrt{x} = \frac{3}{4}\). Thus, the limit of the expression as \(x\) approaches infinity is \(\frac{3}{4}\).

Step 5: Verify and Conclude

Verify the calculations to ensure accuracy. Since there is no given option corresponding to \(\frac{3}{4}\), the correct answer is option D: None of these.

Key Concepts

Binomial Expansion

When dealing with limits and complex expressions, the binomial expansion is a powerful tool to simplify calculations. It allows us to approximate expressions in terms of simpler, more manageable pieces.
In the context of this problem, we have \[ \sqrt{x + \frac{3}{2}\sqrt{x}}. \]For large values of \(x\), the binomial theorem enables us to expand this as:

• \( \sqrt{x} \sqrt{1 + \frac{3}{2} \frac{1}{\sqrt{x}}} \approx \sqrt{x} + \frac{1}{2} \cdot \frac{3}{2} \frac{\sqrt{x}}{\sqrt{x}} = \sqrt{x} + \frac{3}{4}. \)

This simplification is based on the idea that for small \(y\), \((1 + y)^n \approx 1 + ny \). Here, \(y = \frac{3}{2} \frac{1}{\sqrt{x}}\) is very small when \(x\) is large, making the expansion highly accurate.

Square Root Simplification

Simplifying nested or complex square roots is often necessary when evaluating limits or making calculations more tractable. In our problem, the initial expression\[ \sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x} \]seems daunting.
To simplify, we can break down \[ \sqrt{x+\sqrt{x}} \approx \sqrt{x} + \frac{1}{2}\sqrt{x} = \frac{3}{2}\sqrt{x}. \]Thus, we approximate the expression inside the square root as \[ x + \frac{3}{2}\sqrt{x}. \]This step reduces the complexity of the square root and makes further calculations much more manageable. By simplifying square roots this way, we can focus on the larger structures of the problem without losing accuracy in our final approximation.

Infinity Limit Problem

Limits involving infinity often require special strategies because they involve undefined forms. For the given problem, we need to assess what happens to our expression as \(x\) grows indefinitely large.
The expression \[ \lim_{x \to \infty} \left( \sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x} \right) \]is evaluated by focusing on the dominant terms, or the terms which grow fastest as \(x\) approaches infinity.

• By approximating and simplifying, we isolated the expression to \( \frac{3}{4} \), which does not change as \(x\) goes to infinity.

It's important to remember that such evaluations involve approximations which depend heavily on correctly identifying dominant terms and simplifying the expression to a form that can consistently represent large values.