Problem 16 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

\(\lim _{y \rightarrow 0} \frac{\sqrt{1+\sqrt{1+y^{4}}}-\sqrt{2}}{y^{4}} \quad\) [Jan. 9, 2019 (I)] (a) exists and equals \(\frac{1}{4 \sqrt{2}}\) (b) exists and equals \(\frac{1}{2 \sqrt{2}(\sqrt{2}+1)}\) (c) exists and equals \(\frac{1}{2 \sqrt{2}}\) (d) does not exist

Step-by-Step Solution

Answer

The limit exists and equals \( \frac{1}{4\sqrt{2}} \) (Option a).

Step 1: Recognize the Problem Type

The problem requires us to evaluate the limit \( \lim_{y \to 0} \frac{\sqrt{1+\sqrt{1+y^4}}-\sqrt{2}}{y^4} \). Since this is of form 0/0 as \( y \to 0 \), it suggests that we may need to use a series expansion (such as a Taylor series) to find the limit.

Step 2: Simplify the Inner Expression

Consider the expression inside the outer square root, \( \sqrt{1+y^4} \). For small \( y \), we can use a Taylor expansion: \( 1 + y^4 \approx 1 + \frac{y^4}{2} \) when expanded around 0. Thus, \( \sqrt{1+y^4} \approx 1 + \frac{y^4}{2} \).

Step 3: Simplify the Outer Expression

Now, substitute this approximation into the outer square root: \( \sqrt{1 + (1 + \frac{y^4}{2})} \). This simplifies further to \( \sqrt{2 + \frac{y^4}{2}} = \sqrt{2} \sqrt{1 + \frac{y^4}{4}} \).

Step 4: Use Binomial Approximation

Apply binomial expansion to \( \sqrt{1 + \frac{y^4}{4}} \): \( \approx 1 + \frac{1}{8}y^4 \). Therefore, \( \sqrt{2 + \frac{y^4}{2}} \approx \sqrt{2}(1 + \frac{1}{8}y^4) \). Thus, \( \sqrt{2 + \frac{y^4}{2}} \approx \sqrt{2} + \frac{\sqrt{2}}{8} y^4 \).

Step 5: Evaluate Limit

Substitute back into the original limit expression: \[ \lim_{y \to 0} \frac{\sqrt{2} + \frac{\sqrt{2}}{8}y^4 - \sqrt{2}}{y^4} = \lim_{y \to 0} \frac{\frac{\sqrt{2}}{8}y^4}{y^4} = \frac{\sqrt{2}}{8} \].

Step 6: Final Simplification and Answer

Simplify \( \frac{\sqrt{2}}{8} = \frac{1}{4\sqrt{2}} \). This matches option (a). Therefore, the limit exists and equals \( \frac{1}{4\sqrt{2}} \).

Key Concepts

Binomial Expansion

Binomial expansion is a powerful mathematical tool used to simplify expressions raised to a power. It's particularly handy when dealing with expressions like \((1 + x)^n\) where \(n\) is a real number.
The general binomial expansion formula is:

• \((1 + x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \ldots \)

This allows us to approximate values by truncating higher-order terms for small \(x\). In this exercise, we used binomial expansion to approximate \(\sqrt{1 + \frac{y^4}{4}}\) as \(1 + \frac{1}{8}y^4\). This is because for small values of \(y^4\), the higher-order terms become negligible, making calculation simpler.
Remember:

• The binomial expansion works best for expressions where \(x\) is a small number (close to zero).
• We often truncate the expansion after a few terms when precision becomes less impactful.

Taylor Series

The Taylor series is a crucial concept in calculus used to represent functions as infinite sums of terms calculated from its derivatives at a single point. It provides an approximation of a function that can be extremely accurate.
For a function \(f(x)\) about a point \(a\), Taylor series is given by:

• \(f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \ldots\)

In our exercise, Taylor series was used to expand \(\sqrt{1 + y^4}\). For small \(y\), this was approximated as \(1 + \frac{y^4}{2}\). The adaptability of Taylor series in approximating functions makes it a valuable tool in limit evaluation, particularly when the limit leads to an indeterminate form like \(0/0\).
Key points about Taylor series:

• It helps in converting complex expressions into simpler polynomials, facilitating easier calculations.
• It is particularly useful when \(x\) is near \(a\), providing accurate estimates.

Series Expansion

Series expansion is a method used to express a function in terms of a sum of powers of its variable, usually making complex limits and operations more manageable. A series can be finite or infinite based on the requirements of approximation or precision.
In the context of this exercise, series expansions like the Taylor series are used to simplify expressions that otherwise cause indeterminate forms such as \(0/0\).
Types of Series:

• Taylor series: Expands functions around a particular point using derivatives.
• Binomial series: Often used for roots of expressions, particularly when the power is a fraction or irrational.

When dealing with limits, especially when direct substitution leads to indeterminate forms, series expansions are an essential tool. They allow breaking down complex expressions into polynomial approximations, which are easier to manipulate and often resolve indeterminate forms.
Make sure:

• To comprehend the order of expansion needed based on the problem context.
• To choose the right expansion series tailored for the function type and precision required.