Problem 12 - Calculus 1 Course in Mathematics for the IIT JEE and Other Engineering Exams

Question

\(\lim _{x \rightarrow \infty}\left(\sqrt{x^{2}+8 x+3}-\sqrt{x^{2}+4 x+3}\right)\) is equal to (a) 0 (b) \(\infty\) (c) 2 (d) \(1 / 2\)

Step-by-Step Solution

Answer

The limit is 2, so the correct answer is (c) 2.

Step 1: Identify the Limit Expression

The given expression to solve the limit for is \( \lim _{x \rightarrow \infty}\left(\sqrt{x^{2}+8 x+3}-\sqrt{x^{2}+4x+3}\right) \). We need to find the value of this expression as \( x \) approaches infinity.

Step 2: Rationalize the Expression

To simplify the expression, multiply and divide by the conjugate: \( \frac{\left(\sqrt{x^{2}+8x+3} - \sqrt{x^{2}+4x+3}\right)\left(\sqrt{x^{2}+8x+3} + \sqrt{x^{2}+4x+3}\right)}{\sqrt{x^{2}+8x+3} + \sqrt{x^{2}+4x+3}} \). This will help in eliminating the square roots.

Step 3: Simplify the Numerator

The numerator becomes \( (x^2 + 8x + 3) - (x^2 + 4x + 3) \), which simplifies to \( 4x \).

Step 4: Simplify the Denominator

The denominator is \( \sqrt{x^2 + 8x + 3} + \sqrt{x^2 + 4x + 3} \).

Step 5: Factor Out \( x \) from the Denominator

Both terms in the denominator have a dominant term of \( x \). Factor out \( x \) from under the square roots to get: \( x \left(\sqrt{1 + \frac{8}{x} + \frac{3}{x^2}} + \sqrt{1 + \frac{4}{x} + \frac{3}{x^2}}\right) \).

Step 6: Simplify the Limit Expression

So the expression simplifies to \( \frac{4x}{x \left(\sqrt{1 + \frac{8}{x} + \frac{3}{x^2}} + \sqrt{1 + \frac{4}{x} + \frac{3}{x^2}}\right)} \), which simplifies further to \( \frac{4}{\sqrt{1 + \frac{8}{x} + \frac{3}{x^2}} + \sqrt{1 + \frac{4}{x} + \frac{3}{x^2}}} \).

Step 7: Take the Limit as \( x \) Approaches Infinity

As \( x \to \infty \), \( \frac{8}{x} \to 0 \), \( \frac{4}{x} \to 0 \), and \( \frac{3}{x^2} \to 0 \), so the expression becomes \( \frac{4}{\sqrt{1} + \sqrt{1}} \).

Step 8: Calculate the Final Value

The expression finally simplifies to \( \frac{4}{2} = 2 \).

Key Concepts

Limit Evaluation

Understanding limit evaluation is crucial for grasping how functions behave as they approach a certain point or infinity. This concept involves finding the value that a function approaches as the input tends towards a certain number.

For instance, in the example problem, we aim to evaluate the limit of the expression \( \lim _{x \rightarrow \infty}(\sqrt{x^{2}+8 x+3}-\sqrt{x^{2}+4x+3}) \). The key objective is to determine what value this expression closely approaches when \( x \) becomes extremely large.

Here's how you can think about evaluating limits:

• Look for terms that dominate the behavior of the expression. In this case, \( x^2 \) is the dominant term.
• Simplify the expression by removing or reducing the influence of less significant terms.
• Assess the limit by considering the behavior of these dominant terms. As \( x \to \infty \), any terms with \( 1/x \) vanish, making the calculation more straightforward.

Getting the hang of limit evaluation helps you anticipate how functions behave at extremes, a fundamental skill in calculus.

Infinite Limits

Infinite limits describe the behavior of a function as the input approaches infinity. In simpler terms, we're focusing on what happens when \( x \) increases without bound. This concept helps us understand both large-scale and asymptotic behavior of functions.

In our exercise, the function \( \sqrt{x^{2}+8 x+3}-\sqrt{x^{2}+4x+3} \) is analyzed as \( x \rightarrow \infty \). Here, the dominant terms are clear as everything is multiplied or added to \( x^2 \) under the square roots.

• Notice that minor terms become negligible. Thus, terms like \( 8/x \) and \( 4/x \) tend to zero as \( x \) grows.
• The expression simplifies significantly since the leading terms dictate the function's nature at infinite values of \( x \).
• The numerical approach, namely finding bounds or simple expressions, then becomes feasible.

This insight provides guidance in analyzing the function's growth or rate of increase, immensely valuable in calculus and advanced mathematics.

Rationalization of Expressions

Rationalization is a technique used to eliminate roots or radicals in mathematical expressions, making them easier to manage and evaluate, especially in limits. In problems involving subtraction of square roots, like our example, rationalization effectively simplifies the expression.

To rationalize an expression, multiply and divide by the conjugate. Here's how it works in our problem:

• Multiply by the conjugate: \( (\sqrt{x^{2}+8x+3} + \sqrt{x^{2}+4x+3}) \), which is the mirror image in terms of operation signs.
• This operation effectively removes the square roots from the numerator, converting it into a simpler polynomial form.
• Ultimately, this simplifies both the evaluation and limit calculation of the expression.

Rationalizing makes complex expressions simpler to work with and allows the evaluation of limits more straightforward, showcasing a powerful tool in calculus.