Problem 49 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

$$ \lim _{x \rightarrow 0} \frac{x \sqrt[3]{z^{2}-(z-x)^{2}}}{\left(\sqrt[3]{8 x z-4 x^{2}}+\sqrt[3]{8 x z}\right)^{4}} \text { is equal to } $$ (A) \(\frac{z}{2^{11 / 3}}\) (B) \(\frac{1}{2^{23 / 3} \cdot z}\) (C) \(2^{21 / 3} z\) (D) None of these

Step-by-Step Solution

Answer

The correct answer is (D) None of these.

Step 1: Simplifying the Numerator

The numerator of the limit is \( x \sqrt[3]{z^2 - (z-x)^2} \). Notice that inside the cube root, \( z^2 - (z-x)^2 \) simplifies to \( z^2 - (z^2 - 2zx + x^2) = 2zx - x^2 \). Thus, the numerator becomes \( x \sqrt[3]{2zx - x^2} \).

Step 2: Simplifying the Denominator

The denominator involves two cube roots, \( \sqrt[3]{8xz - 4x^2} + \sqrt[3]{8xz} \). When \( x \) approaches 0, both expressions can be simplified.1. \( \sqrt[3]{8xz - 4x^2} \) approaches \( \sqrt[3]{8xz} \) since the term \( 4x^2 \) becomes negligible.2. \( \sqrt[3]{8xz} \) is exactly itself.Thus, the denominator becomes \((\sqrt[3]{8xz} + \sqrt[3]{8xz})^4 = (2 \cdot \sqrt[3]{8xz})^4\).

Step 3: Simplify the Expression

Now, simplify the limit expression:\[ \frac{x \sqrt[3]{2zx - x^2}}{(2 \cdot \sqrt[3]{8xz})^4} \]This becomes:1. Substitute and simplify the denominator as \((2 \cdot 2^{1/3} (xz)^{1/3})^4 = 2^4 \cdot 2^{4/3} \cdot (xz)^{4/3}\).2. Now substitute this back:\[ \frac{x (2zx - x^2)^{1/3}}{2^{16/3} (xz)^{4/3}} \].

Step 4: Evaluate the Limit as x Approaches 0

Since \((2zx - x^2)\) can be approximated by \(2zx\) when \(x\) approaches 0, the numerator \(x (2zx)^{1/3} = x (2^{1/3} z^{1/3} x^{1/3}) = 2^{1/3} x^{4/3} z^{1/3} \).Substitute this back to the equation:\[ \frac{2^{1/3} x^{4/3} z^{1/3}}{2^{16/3} x^{4/3} z^{4/3}} = \frac{2^{1/3}}{2^{16/3} z} \].

Step 5: Simplify and Identify the Correct Answer

Simplify the expression further:\[ \frac{2^{1/3}}{2^{16/3} z} = \frac{1}{2^{15/3} \cdot z} = \frac{1}{2^5 \cdot z} = \frac{1}{32z} \].The given answer choices do not exactly match \(\frac{1}{32z}\), therefore the answer is (D) None of these.

Key Concepts

Cube Roots

When we talk about cube roots, we refer to finding a number that, when multiplied by itself three times, results in the original number we started with. Imagine having a cube with equal edges – the cube root is the length of each edge if you know the volume of the cube. Mathematically, for a given number \( a \), the cube root is represented as \( \sqrt[3]{a} \).
Cube roots are slightly more complex to deal with than square roots primarily because they involve three multiplications instead of two. In the context of limits, dealing with cube roots typically requires simplifying expressions so you can evaluate how they behave as a variable approaches a specific value.
For example, in the problem given, expressions like \( \sqrt[3]{2zx - x^2} \) and \( \sqrt[3]{8xz} \) are refined by focusing on essential terms while neglecting negligible terms as \( x \) approaches zero.

Simplifying Expressions

Simplifying expressions is a crucial part of calculus, especially when working with complex functions as you prepare to evaluate limits. It involves reducing expressions to their simplest form by factoring, combining like terms, or identifying terms that can be ignored in certain mathematical limits.
In the problem, simplifying occurs both in the numerator and the denominator. For the numerator, the expression inside the cube root is simplified from \( z^2 - (z-x)^2 \) to \( 2zx - x^2 \). Simplifying this expression makes it easier to evaluate as \( x \) approaches zero.
In the denominator, cube roots like \( \sqrt[3]{8xz - 4x^2} \) are approached by simplifying it to \( \sqrt[3]{8xz} \), further continuing the simplification to \( (2 \cdot \sqrt[3]{8xz})^4 \). This involves an understanding of how terms behave as a variable tends to a limit, often allowing smaller terms to be effectively ignored.

Evaluating Limits

Evaluating limits is a fundamental concept in calculus, which explores what happens to a function as its variable approaches a particular value. This process involves breaking down complex expressions into simpler forms so that their behavior can be clearly analyzed.
In our example, we aim to find the limit as \( x \) approaches zero for a given fraction. This involves knowing how to handle cube roots and powers effectively, especially when terms can simplify significantly.
Once simplified, the expression \( \frac{2^{1/3} x^{4/3} z^{1/3}}{2^{16/3} x^{4/3} z^{4/3}} \) is evaluated by observing the behavior of different terms as \( x \) goes to zero. Here, essential terms are isolated, and those diminishing disappear, allowing the final evaluation to be achieved. The initial solution identified \( \frac{1}{32z} \) as the result, showing neither match to the suggested options, guiding us to the notion that no correct multiple choice is provided other than the option 'None of these.'
Recognizing such outcomes in evaluation helps build strong calculus skills, essential in higher mathematics and real-world problem solving.