Problem 11 - Calculus 1 Course in Mathematics for the IIT JEE and Other Engineering Exams

Question

The value of \(\lim _{x \rightarrow \infty}\left(\frac{x^{2}+b x+4}{x^{2}+a x+5}\right)\) is (a) \(b / a\) (b) 1 (c) 0 (d) \(4 / 5\)

Step-by-Step Solution

Answer

The value of the limit is 1 (option b).

Step 1 - Identify the Dominant Terms

When evaluating the limit as \(x\) approaches infinity, the dominant terms in the numerator and denominator are those with the highest powers. In this case, \(x^2\) is the dominant term in both the numerator \(x^2 + bx + 4\) and the denominator \(x^2 + ax + 5\).

Step 2 - Simplify the Limit Expression

Divide every term in both the numerator and the denominator by the dominant term, \(x^2\). This gives us: \(\lim_{x \to \infty} \left(\frac{x^2/x^2 + bx/x^2 + 4/x^2}{x^2/x^2 + ax/x^2 + 5/x^2}\right)\).

Step 3 - Evaluate the Resulting Expression

Simplify the expression: \(\lim_{x \to \infty} \left(\frac{1 + b/x + 4/x^2}{1 + a/x + 5/x^2}\right)\). As \(x\) approaches infinity, \(b/x\), \(4/x^2\), \(a/x\), and \(5/x^2\) all approach zero.

Step 4 - Conclude the Limit Value

With the higher degree terms approaching zero, the fraction simplifies to \(\frac{1}{1}\). Therefore, the limit is 1.

Key Concepts

Dominant Terms Method

When dealing with limits, especially as variables approach infinity, the 1dominant terms method d is like focusing on the loudest in a crowd - the terms with the highest power. In polynomial expressions, these terms overshadow all others as the variable grows larger.

For example, consider the expression c\(x^2 + bx + 4\) d. As \(x\) becomes very large, \(x^2\) becomes much more significant than any \(bx\) or constant like 4. This means \(x^2\) is the dominant term. Similarly, in the denominator \(x^2 + ax + 5\), \(x^2\) is also the dominant term.

• This method simplifies the process of evaluating limits.
• Focuses only on terms that actually affect the limit at infinity.

By concentrating on these dominant terms, you can ignore smaller terms that shrink away as the variable grows, making it easier to handle complicated expressions. This forms the basis for quick and efficient calculations of limits at infinity.

Evaluating Limits at Infinity

As you tackle limits where \(x\) approaches infinity, it's crucial to simplify and observe the behavior of the expression in such conditions.

First, by identifying dominant terms, you divide each term by the highest power variable. Using our example, each term in the numerator and denominator is divided by \(x^2\), the dominant power: \[\lim_{x \to \infty} \left(\frac{x^2/x^2 + bx/x^2 + 4/x^2}{x^2/x^2 + ax/x^2 + 5/x^2}\right)\].

This results in \[1 + \frac{b}{x} + \frac{4}{x^2}\text{ over } 1 + \frac{a}{x} + \frac{5}{x^2}\].

Now, as \(x\) approaches infinity:

• \(\frac{b}{x} \rightarrow 0\)
• \(\frac{4}{x^2} \rightarrow 0\)
• \(\frac{a}{x} \rightarrow 0\)
• \(\frac{5}{x^2} \rightarrow 0\)

All fractional components approach zero, simplifying the expression to \(\frac{1}{1}\).

This showcases how dominating terms dictate the behavior at infinity, proving that simpler expressions often determine the outcome in such scenarios.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing an expression to its simplest form. When evaluating limits, particularly at infinity, this step is essential. It allows us to focus on the core behavior of the function.

Consider \[\frac{x^2 + bx + 4}{x^2 + ax + 5}\].

We began by dividing every term by \(x^2\), the dominant term.

This transforms it to \[\frac{1 + b/x + 4/x^2}{1 + a/x + 5/x^2}\], which simplifies to \(1\), as discussed earlier.

Here's why simplifying is beneficial:

• It minimizes complex calculations.
• Provides a clearer view of how functions behave as variables change significantly.
• Makes it easier to predict outcomes, like limits, quickly.

By simplifying expressions before evaluating, we ensure efficiency and accuracy. This empowers us to handle more complex calculus problems with ease.