Problem 82 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(y=x+\frac{\sqrt{x}}{x+\frac{\sqrt{x}}{x+\frac{\sqrt{x}}{\ldots \infty}}}\), then \(\lim _{x \rightarrow \infty} \frac{x}{y}\) is equal to (A) 1 (B) \(-1\) (C) 0 (D) None of these

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Understanding the Given Expression

The expression for \(y\) is given as \(y = x + \frac{\sqrt{x}}{x + \frac{\sqrt{x}}{x + \frac{\sqrt{x}}{\ldots \infty}}}\). This is an infinite continued fraction. Note that due to the structure, \(y\) satisfies \(y = x + \frac{\sqrt{x}}{y}\).

Step 2: Simplifying the Expression

Assume \( y = x + \frac{\sqrt{x}}{y} \). Multiply both sides by \(y\) to obtain \( y^2 = xy + \sqrt{x} \). Rearranging gives the quadratic equation \( y^2 - xy - \sqrt{x} = 0 \).

Step 3: Solving the Quadratic Equation

Use the quadratic formula, \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), with \(a = 1\), \(b = -x\), and \(c = -\sqrt{x}\). This gives \( y = \frac{x \pm \sqrt{x^2 + 4\sqrt{x}}}{2} \).

Step 4: Approximating for Large \(x\)

For \(x\) approaching infinity, simplify \( y \approx \frac{x + x}{2} = x \) since the \( 4\sqrt{x} \) term becomes negligible compared to \(x^2\). Thus, \( y \approx x \).

Step 5: Calculating the Limit

Evaluate \( \lim_{x \to \infty} \frac{x}{y} \). Since \(y \approx x\) for large \(x\), \( \frac{x}{y} \approx \frac{x}{x} = 1 \).

Step 6: Conclusion

The limit \( \lim_{x \to \infty} \frac{x}{y} \) is equal to 1.

Key Concepts

Infinite Continued Fractions

Infinite continued fractions are expressions that extend indefinitely. In this problem, the given expression for \( y \) is a perfect example: \( y=x+\frac{\sqrt{x}}{x+\frac{\sqrt{x}}{x+\frac{\sqrt{x}}{\ldots \infty}}} \). These fractions continue endlessly, which can initially seem complex, but they follow a repetitive pattern.
To approach them, it's often helpful to define a part of the sequence as an unknown, like how \( y \) satisfies the condition \( y = x + \frac{\sqrt{x}}{y} \). This can turn an infinite problem into something more manageable. By isolating and working with such equations, you reduce complexity and set the stage for solving them through simplification and solution of equations. This technique is vital in various advanced mathematics fields.

Quadratic Equations

A quadratic equation is any equation that can be arranged in the form \( ax^2 + bx + c = 0 \). Here, we see a quadratic emerge from the infinite continued fraction: \( y^2 = xy + \sqrt{x} \) simplifies and rearranges to \( y^2 - xy - \sqrt{x} = 0 \).
To solve this, apply the quadratic formula: \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). In our case, \( a = 1 \), \( b = -x \), and \( c = -\sqrt{x} \). Solving with these values gives us \( y = \frac{x \pm \sqrt{x^2 + 4\sqrt{x}}}{2} \). Quadratic equations like this are foundational in algebra and allow us to find precise values where simple equations turn complex.

Asymptotic Approximation

Asymptotic approximation is a method used in calculus and mathematical analysis to approximate functions as variables within them approach certain limits. This technique becomes a handy tool when dealing with complex equations, such as those involving infinite continued fractions. In this exercise, as \( x \) approaches infinity, our solution \( y \approx x \) simplifies considerably.
Asymptotic analysis involves recognizing which parts of an equation become negligible when compared to others. For instance, as \( x \to \infty \), \( x^2 + 4\sqrt{x} \) simplifies to just \( x^2 \) because the term \( 4\sqrt{x} \) becomes insignificant in comparison. This helps simplify equations, making it easier to determine limits or behavior of functions without exhaustive calculations.

Mathematics Problem Solving

Mathematics problem solving requires a systematic approach to find solutions. It starts with understanding the problem, identifying patterns or structures, and applying appropriate mathematical principles. In our exercise, breaking down an infinite continued fraction into a more digestible quadratic equation was a key problem-solving step.
Effective problem-solving strategies often involve:

• Understanding the problem through simplification or reformulation.
• Utilizing known mathematical theories like quadratic formulae.
• Applying approximation techniques for tackling complex limits.

Ultimately, the goal is to enhance clarity and achieve solutions efficiently. Practicing these strategies builds strong analytical skills and boosts confidence in mathematical understanding.