Problem 95 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \lim _{x \rightarrow \infty} \frac{x^{3}+x}{x^{4}-3 x^{2}+1}\\{\text { Ans. } 0\\} $$

Step-by-Step Solution

Answer

The short answer for the limit of the function \(\frac{x^3+x}{x^4-3x^2+1}\) as \(x \rightarrow \infty\) is 0. This is determined by dividing the numerator and denominator by the highest power of x in the denominator, simplifying the fractions, and evaluating the limit as x approaches infinity. The result is \(\frac{0}{1}\), which simplifies to 0.

Step 1: Divide Numerator and Denominator by Highest Power of x in Denominator

We have to find the limit of the function \(\frac{x^3+x}{x^4-3x^2+1}\) as \(x \rightarrow \infty\). First, divide both the numerator and denominator by \(x^4\): \[ \lim_{x \rightarrow \infty} \frac{\frac{x^3+x}{x^4}}{\frac{x^4-3x^2+1}{x^4}} \]

Step 2: Simplify the Fractions

Now, simplify the fractions: \[ \lim_{x \rightarrow \infty} \frac{\frac{x^3}{x^4}+\frac{x}{x^4}}{\frac{x^4}{x^4}-\frac{3x^2}{x^4}+\frac{1}{x^4}} \\ \lim_{x \rightarrow \infty} \frac{\frac{1}{x}+\frac{1}{x^3}}{1-\frac{3}{x^2}+\frac{1}{x^4}} \]

Step 3: Evaluate the Limit as x Approaches Infinity

As \(x \rightarrow \infty\), all the terms in the numerator and denominator with powers of x in the denominator will approach 0: \[ \lim_{x \rightarrow \infty} \frac{\frac{1}{x}+\frac{1}{x^3}}{1-\frac{3}{x^2}+\frac{1}{x^4}} = \frac{0+0}{1-0+0} = \frac{0}{1} \]

Step 4: Express the Final Answer

The limit of the function \(\frac{x^3+x}{x^4-3x^2+1}\) as \(x \rightarrow \infty\) is 0: \[ \lim_{x \rightarrow \infty} \frac{x^3+x}{x^4-3x^2+1} = 0 \]

Key Concepts

Infinity in Calculus

In calculus, the concept of infinity is crucial for understanding how functions behave as variables grow very large or very small. When we say \(x \rightarrow \infty\), we are interested in what happens to the function as \(x\) tends toward very large values. This often helps us identify horizontal asymptotes or limits of functions that extend towards infinity.

For instance, when dealing with rational functions, as seen in the original exercise, the behavior of the function as \(x \to \infty\) shows us how the values of infinitesimally minor terms become negligible. This means terms like \(\frac{1}{x}\) or \(\frac{1}{x^3}\) tend towards zero, letting us simplify the function.

Infinity in calculus doesn't refer to a number but rather a directional approach to limits. It provides an understanding of how functions can stretch indefinitely in their respective ranges, creating graphs with infinite tails or widths.

Simplifying Rational Functions

Rational functions consist of a numerator and denominator, both of which are polynomials, like \(f(x) = \frac{P(x)}{Q(x)}\). Simplifying these functions involves making the function easier to work with by eliminating common factors or dividing by the highest power of the variable when assessing limits.

In the given exercise, we simplified the rational function \(\frac{x^3+x}{x^4-3x^2+1}\) by dividing every term by the highest power in the denominator, which is \(x^4\).

• Each reduced term like \(\frac{x^3}{x^4} = \frac{1}{x}\) becomes smaller as \(x\) increases, simplifying the evaluation.
• This results in the fractions \(\frac{1}{x}\), \(\frac{1}{x^3}\) and others, which approach zero, leaving a much simpler form to evaluate.

The aim of this process is to reduce complex expressions into more manageable forms, especially when trying to find limits as variables approach infinity.

Power Functions in Calculus

Power functions are functions of the form \(f(x) = x^n\), where \(n\) is a real number. These functions are straightforward and involve raising a variable to a constant power. In calculus, understanding power functions is key to grasping more complex polynomial behavior.

In our exercise, breaking down the power relationship within the fraction \(\frac{x^3+x}{x^4-3x^2+1}\) showed how different powers of \(x\) interact. Since \(x\) as a power function dominates when compared to constants or lower powers of \(x\), higher power terms in the denominator like \(x^4\) have more impact than those in the numerator like \(x^3\).

• The simplification process shows us how to prioritize and isolate these power functions for limit evaluation.
• Understanding this helps in identifying which terms become insignificant as they approach zero, leading to much simpler evaluations.

Through this, we not only evaluate limits proficiently but also appreciate the hierarchical nature of power functions in calculus.