Problem 442 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { If } \lim _{x \rightarrow \infty} \frac{x^{3}+1}{x^{2}+1}-a x-b=2 \text { , then find } a \text { and } b \text { . } $$

Step-by-Step Solution

Answer

The values of \(a\) and \(b\) are 1 and -2 respectively.

Step 1: Simplify the equation

Rewrite the equation as \( \lim _{x \rightarrow \infty} [\frac{x^{3}+1}{x^{2}+1} - (ax + b)] - 2 = 0 \) Then simplify the equation by considering \(x\) tends to infinity: \[\lim _{x \rightarrow \infty} [\frac{x(x^2 + \frac{1}{x^2})}{x^2 + \frac{1}{x^2}} - (ax + b)] - 2 = 0\] Simplify the terms within the limit: \[\lim _{x \rightarrow \infty} [x - (ax + b)] - 2 = 0\] Finally rewrite it as \( (1-a)x - b -2 = 0 \)

Step 2: Apply the limit as \(x\) tends to infinity

We have \( (1-a)x - b - 2 = 0 \) Since \(x\) tends to infinity, the term \((1 - a)x\) will also tend to infinity unless \(a = 1\), such that the limit exists. By setting \(a = 1\), the equation simplifies to \( -b - 2 = 0 \)

Step 3: Solve for \(b\)

From the equation \( -b - 2 = 0 \), we can solve for \(b\) by adding 2 to both sides of the equation, thus we get \(b = -2\)

Key Concepts

Infinite Limits

Infinite limits occur when the variable approaches infinity or negative infinity. In this exercise, we look at the behavior of the function as the value of \( x \) becomes very large. The equation given requests the limit as \( x \) approaches infinity. This means we are interested in how the expression \( \frac{x^3+1}{x^2+1} - ax - b \) behaves as \( x \) grows larger and larger.

This can initially seem daunting, but there's a systematic way to tackle it. As \( x \) tends to infinity, the highest power of \( x \) generally dominates the behavior of a polynomial. In such cases, it's helpful to simplify complex expressions by focusing on the terms with the largest powers and seeing how they affect the outcome as \( x \) tends towards infinity.

Asymptotic Behavior

Asymptotic behavior deals with how functions behave as they approach a certain limit. In this scenario, as \( x \) approaches infinity, the expression \( \frac{x^3+1}{x^2+1} \) can be simplified to \( x \), using asymptotic analysis. Asymptotic analysis means zooming in on the leading terms of an equation to predict the behavior at the extremes.

Here, \( x^3 \) dominates the numerator and \( x^2 \) dominates the denominator. Thus, when simplified, the expression tends to behave like \( x \). We used this analysis to reconcile the expression \( \frac{x^3+1}{x^2+1} \) by introducing \( \frac{1}{x^2} \) terms.

With this simplified understanding of asymptotic behavior, we're able to set the equation \( (1-a)x-b-2=0 \), leading to the necessity for \( a=1 \) so that there is no infinite behavior.

Limit Algebra

Limit algebra involves the rules and strategies we use to manipulate limits mathematically. One key aspect of limit algebra is the recognition of the importance of balancing terms so that the infinite behavior of functions can be controlled.

In this exercise, to find meaningful values for \( a \) and \( b \), we manipulated the polynomial by equating and simplifying it to ensure that no infinite terms remained. This manipulation allowed us to apply the rule where coefficients of like terms in the polynomial must match set values, particularly when dealing with terms like \( x \) and \( b \).

• Ensuring the coefficient of \( x \) equals zero when equaling to a constant, results in \( a = 1 \).
• Similarly, balancing the constant terms results in \( b = -2 \).

These steps highlight how essential it is to meticulously perform algebraic operations in limits to uncover important characteristics of functions as they integrate with infinity.