Problem 316 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \lim _{x \rightarrow \infty} \sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x} .\left\\{\text { Ans. } \frac{1}{2}\right\\} $$

Step-by-Step Solution

Answer

In summary, the limit as \(x\) approaches infinity of the given expression, \(\lim _{x \rightarrow \infty} \sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x}\), does not exist. The original given answer, \(\frac{1}{2}\), was incorrect.

Step 1: Rewrite the expression with a common factor

We are given the limit: \(\lim _{x \rightarrow \infty} \sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x}\) Notice that there is a common factor of x within each square root. To simplify this expression, rewrite it with a common factor: \(\lim _{x \rightarrow \infty} \sqrt{x\left(1+\sqrt{1+\frac{1}{\sqrt{x}}}\right)}-\sqrt{x}\)

Step 2: Factor out the common factor from the square root

Now we can factor out the common factor x from the square root: \(\lim _{x \rightarrow \infty} \sqrt{x}\left(\sqrt{1+\sqrt{1+\frac{1}{\sqrt{x}}}}-1\right)\)

Step 3: Evaluate the limit by considering the behavior when x approaches infinity

As x approaches infinity, we consider the behavior of the inner square root: \(\lim _{x \rightarrow \infty} \sqrt{1+\frac{1}{\sqrt{x}}} = \sqrt{1+0} = \sqrt{1}\) Thus, our expression becomes: \(\lim _{x \rightarrow \infty} \sqrt{x}\left(\sqrt{1+\sqrt{1}}-1\right)\) Now, we have: \(\lim _{x \rightarrow \infty} \sqrt{x}\left(\sqrt{1+1}-1\right) = \lim _{x \rightarrow \infty} \sqrt{x}\left(\sqrt{2}-1\right) = \lim _{x \rightarrow \infty} \frac{(\sqrt{2}-1)\sqrt{x}}{1}\) As x approaches infinity, the fraction \(\frac{(\sqrt{2}-1)\sqrt{x}}{1}\) also approaches infinity, making the limit go to infinity.

Step 4: Simplify and evaluate the final result

The expression approaches infinity as x approaches infinity, and thus the limit does not exist. The answer given was mistaken, as it mentioned the answer was \(\frac{1}{2}\). The correct answer should be: The limit does not exist as x approaches infinity.

Key Concepts

Infinity Limits

Understanding limits as a function approaches infinity can sometimes be tricky but also fascinating. In this concept, we look at how a function behaves as the variable within it grows infinitely larger. When dealing with expressions that involve limits approaching infinity, particularly with square roots, we notice specific behaviors like within the given problem. Here, we restructure the internal expression to make the limit more apparent, often simplifying the algebra to manage seemingly undefined forms. This practice is known as finding the infinity limits, which allows us to analyze behavior without actually reaching infinity. It's crucial because it offers insights into the asymptotic behaviors of functions, a vital part of calculus.

Square Roots

Square roots can sometimes complicate limit problems, but they also offer a unique lens through which to simplify and solve them. When dealing with expressions including square roots, especially with functions approaching infinity, the main task is often to simplify these parts of the expression. In our example, the operation of taking the square root is performed multiple times within the equation. We simplify by identifying common factors and dealing with the nested layers of roots to ease the computations. The property that \(\sqrt{x}\cdot\sqrt{y}=\sqrt{xy}\) helps in factorizing out common terms like we do with \(\sqrt{x(1+...) - \sqrt{x}\), making it simpler to focus on the critical components of the function as \(x\) grows very large.

Factorization

Factorization becomes an essential tool when simplifying complex expressions, especially those involving square roots within limit problems. Through factorization, we can break down intricate algebraic expressions into simpler, more manageable parts. This process often reveals hidden constants or reduces the complexity by canceling common terms. By factoring out terms, particularly when encountering nested roots, we simplify the expression radically. For instance, in our step-by-step solution, this allowed the initial complex expression to reduce significantly, eventually simplifying the behavior of the function as it approaches infinity. Thus, factorization isn't just an algebraic trick; it’s a powerful method to simplify and solve problems dealing with infinite values.