Problem 48 - A Complete Resource Book in Mathematics for JEE Main 2017

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

The limit is (A) \(\frac{2}{\sqrt{3}}\).

Step 1: Identify the Dominant Terms

To evaluate the limit as \(x\) approaches infinity, first identify the dominant terms in both the numerator and the denominator. The dominant term in the numerator is \(2\sqrt{x}\), and in the denominator, it is \(\sqrt{3x}\). This is because these terms grow the fastest as \(x\) becomes very large.

Step 2: Simplify the Expression

Divide every term in the numerator and the denominator by \(\sqrt{x}\), the dominant term identified. The expression becomes \[\frac{2 + \frac{3}{x^{1/6}} + \frac{5}{x^{2/5}}}{\sqrt{3} - \frac{2}{3x} + \frac{1}{x^{2/3}}}.\]

Step 3: Evaluate the Limit of Each Term

As \(x\) approaches infinity, the fractions \(\frac{3}{x^{1/6}}\), \(\frac{5}{x^{2/5}}\), \(\frac{2}{3x}\), and \(\frac{1}{x^{2/3}}\) all approach zero. Therefore, the expression simplifies to \[\frac{2}{\sqrt{3}}.\]

Step 4: Conclusion and Final Result

Since the limit evaluation shows that all other terms vanish as \(x\) approaches infinity, the limit of the whole expression is \(\frac{2}{\sqrt{3}}\).

Key Concepts

Limits

Limits in calculus help us understand the behavior of functions as they approach certain points or infinity. In this exercise, we care about the limit of a function as the variable \(x\) heads towards infinity. When dealing with limits, the key is to determine what happens to the function values.

• Do they get closer to a specific value?
• Do they grow without bound?

Understanding limits is crucial because it lays the foundation for more advanced topics, such as derivatives and integrals. Limits also help predict the "end behavior" of functions. And when the input values get very large or very small, this end behavior could tell us a lot about the function.

Infinite Limits

Infinite limits specifically deal with what happens to a function's values as \(x\) moves toward positive or negative infinity. In this exercise, we're considering the limit as \(x\) approaches positive infinity. This means we're interested in how the function behaves when \(x\) gets really, really large.When addressing infinite limits, certain terms in the function will dominate, becoming significant as \(x\) increases. Others become negligible. This dynamic is because some terms grow much faster than others. Focusing on the dominant terms allows us to simplify the problem. For a clearer understanding, let's explore dominant terms more closely in the next section.

Dominant Terms

Dominant terms in a function are the ones that have the most significant effect when \(x\) becomes very large. In this fraction, the dominant term in the numerator \(2\sqrt{x}\) grows faster than the others because of its power. Similarly, \(\sqrt{3x}\) in the denominator plays a pivotal role. By dividing all parts of the fraction by these dominant terms, we simplify the expression to an easier form, making it manageable to evaluate. This process effectively narrows down the equation to what's essential, showing that as \(x\) approaches infinity, other terms fade into insignificance, allowing us to determine the function's limit precisely.