Problem 13 - Precalculus Mathematics for Calculus

Question

Find the limit. $$\lim _{x \rightarrow \infty} \frac{x^{4}}{1-x^{2}+x^{3}}$$

Step-by-Step Solution

Answer

The limit is infinity.

Step 1: Identify the highest power of x in the denominator

In the function \( \frac{x^4}{1 - x^2 + x^3} \), the denominator has terms \(1\), \(-x^2\), and \(x^3\). The highest power of \(x\) in the denominator is \(x^3\).

Step 2: Divide numerator and denominator by the highest power of x found

Divide every term in the numerator and the denominator by \(x^3\). This gives us \(\frac{x^4/x^3}{1/x^3 - x^2/x^3 + x^3/x^3}\). This simplifies to \(\frac{x}{1/x^3 - 1/x + 1}\).

Step 3: Simplify the expression

After dividing, the expression becomes \(\frac{x}{1/x^3 - 1/x + 1}\). Notice that as \(x \to \infty\), \(1/x\) and \(1/x^3\) both approach 0. Hence, the expression simplifies to \(\frac{x}{0 + 0 + 1}\) or just \(x\).

Step 4: Evaluate the limit as x approaches infinity

Now, we evaluate \(\lim_{x \to \infty} x\). As \(x\) goes to infinity, the limit of \(x\) is infinity.

Key Concepts

Limits

Limits are a fundamental concept in calculus, helping us understand the behavior of a function as its input approaches a particular value or even infinity. In this context, the limit was used to evaluate the behavior of a rational function as the variable approaches infinity.
When evaluating limits, especially as they approach infinity, focus on the terms with the highest exponents, as these dominate the behavior of the function. Smaller terms tend to disappear when compared to larger powers as the variable grows, simplifying the process. Understanding limits is crucial in calculating the behavior at such extremes, ensuring we capture the function's trend accurately.
In our exercise, we worked on a rational function where the terms involving powers of \( x \) were simplified, enabling us to evaluate the limit of \( x \) as it heads towards infinity.

Rational Functions

Rational functions are quotients of two polynomials. In our problem, we were given a rational function \( \frac{x^4}{1-x^2+x^3} \). Such functions are interesting because their behaviors can be complex due to the interactions between the numerator and denominator.
To simplify a rational function when finding its limit, especially for limits approaching infinity, it is often useful to divide both the numerator and the denominator by the highest power of \( x \) found in the denominator. This step comes from the need to reduce the expression's complexity while analyzing the dominant terms, making it easier to deduce the function's long-term behavior.
Understanding these functions’ behaviors is key to solving calculus problems, as they appear frequently in various fields like physics and engineering.

Infinity

Infinity in mathematics is not a number, but a concept representing unboundedness. When working with limits in calculus, infinity is used to describe the behavior of functions as their inputs grow without bound.
In our exercise, we looked at the limit of a rational function as \( x \to \infty \). This involves determining what happens to the function's value when \( x \) becomes extremely large. The function becomes dominated by terms with the highest powers of \( x \), leading us to conclude that the function itself tends towards infinity.
This concept is vital in predicting and understanding mathematical models' behavior that interact with very large values or spaces, ensuring accurate descriptions of growth without constraints.