Problem 48 - A Complete Resource Book in Mathematics for JEE Main

Question

The value of \(\lim _{x \rightarrow \infty} \frac{2 \sqrt{x}+3 \sqrt[3]{x}+5 \sqrt[5]{x}}{\sqrt{3 x-2}+\sqrt[3]{2 x-3}}\) is (A) \(\frac{2}{\sqrt{3}}\) (B) \(\sqrt{3}\) (C) \(\frac{1}{\sqrt{3}}\) (D) None of these

Step-by-Step Solution

Answer

(A) \(\frac{2}{\sqrt{3}}\)

Step 1: Understanding the Overall Limit

We are tasked to find \( \lim_{x \to \infty} \frac{2 \sqrt{x} + 3 \sqrt[3]{x} + 5 \sqrt[5]{x} }{ \sqrt{3x-2} + \sqrt[3]{2x-3} } \). This expression is in an indeterminate form as \( x \to \infty \). To resolve this, we first analyze the highest power of \( x \) in the numerator and the denominator.

Step 2: Analyzing the Dominant Term in the Numerator

In the numerator \( 2 \sqrt{x} + 3 \sqrt[3]{x} + 5 \sqrt[5]{x} \), the dominant term as \( x \to \infty \) is \( 2 \sqrt{x} \) because its power, \( x^{1/2} \), is greater than the others.

Step 3: Analyzing the Dominant Term in the Denominator

In the denominator \( \sqrt{3x-2} + \sqrt[3]{2x-3} \), the dominant term is \( \sqrt{3x-2} \), which simplifies to approximately \( \sqrt{3x} \) for large \( x \).

Step 4: Simplify using Dominant Terms

We simplify the limit by dividing both the numerator and the denominator by the dominant terms found in Steps 2 and 3. This gives us \[ \lim_{x \to \infty} \frac{2 \sqrt{x}}{\sqrt{3x}} \].

Step 5: Simplify Further

Further simplification using properties of limits gives \[ \lim_{x \to \infty} \frac{2}{\sqrt{3}} \cdot \sqrt{\frac{x}{x}} = \frac{2}{\sqrt{3}}. \] This simplification arises because \( \sqrt{\frac{x}{x}} = \sqrt{1} = 1 \).

Step 6: Conclude with the Answer

Therefore, based on our simplifications, the limit evaluates to \( \frac{2}{\sqrt{3}} \). This matches choice (A).

Key Concepts

Indeterminate Forms

In calculus, an indeterminate form arises when there's an ambiguity in the behavior of a function, especially as it approaches a certain point. In our case, as we evaluate \( \lim_{x \rightarrow \infty} \), we encounter expressions like \( \frac{\infty}{\infty} \), where both the numerator and the denominator grow indefinitely. This situation is known as an indeterminate form. These are special since we cannot easily evaluate them without further analysis.

To resolve such forms, we often need to perform additional steps like simplification or applying specific theorems, such as L'Hôpital's rule. However, in our problem, simplification through identification of dominant terms suffices, which lets us uncover the true behavior of the function as \( x \) becomes extremely large.

Simplification of Rational Expressions

Simplifying a rational expression is a crucial step to resolving indeterminate forms, especially when limits are involved. Here, this means strategically rewriting the expression to focus on the most significant terms. The goal is to reduce complexity and isolate terms that control the behavior of both the numerator and denominator as \( x \to \infty \).

In the given exercise, the original expression appears complex, consisting of multiple terms with roots in both the numerator and denominator:

• Numerator: \( 2 \sqrt{x} + 3 \sqrt[3]{x} + 5 \sqrt[5]{x} \)
• Denominator: \( \sqrt{3x-2} + \sqrt[3]{2x-3} \)

Recognizing which terms are dominant (i.e., those with the highest powers of \( x \)) is the key to simplification here. This allows us to simplify the complex rational expression and eventually resolve the limit correctly.

Dominant Term Analysis

Dominant term analysis is a technique used to simplify expressions by identifying which terms most significantly influence the behavior of the expression as a certain variable changes. In the context of limits, especially as \( x \rightarrow \infty \), identifying dominant terms helps reduce the expression to its simplest form.

For our problem, we compare terms in both the numerator and the denominator:

• In the numerator \( (2 \sqrt{x} + 3 \sqrt[3]{x} + 5 \sqrt[5]{x}) \), \( 2 \sqrt{x} \) is dominant because \( x^{1/2} \) grows faster than \( x^{1/3} \) and \( x^{1/5} \).
• In the denominator \( (\sqrt{3x-2} + \sqrt[3]{2x-3}) \), \( \sqrt{3x} \) approximates the dominant behavior of \( \sqrt{3x-2} \) as \( x \to \infty \).

After identifying these terms, we perform the simplification by dividing both the top and the bottom by these dominant terms, ultimately simplifying the entire expression. This ensures the limit can be evaluated accurately and succinctly, resolving the initially indeterminate form.