Question

In Exercises 9 and \(10,\) find \(\lim _{x \rightarrow \infty} h(x),\) if possible. \(f(x)=5 x^{3}-3 x^{2}+10\) (a) \(h(x)=\frac{f(x)}{x^{2}}\) (b) \(h(x)=\frac{f(x)}{x^{3}}\) (c) \(h(x)=\frac{f(x)}{x^{4}}\)

Short answer

The limit as \(x\) approaches infinity for each variation of \(h(x)\) is as follows: (a) \(\lim _{x \rightarrow \infty} h(x) = 5x - 3\), (b) \(\lim _{x \rightarrow \infty} h(x) = 5\), (c) \(\lim _{x \rightarrow \infty} h(x) = 0\)

Step-by-step solution

Find the limit of \(h(x) = \frac{f(x)}{x^{2}}\)

First, divide the numerator of \(h(x)\) by \(x^{2}\). The function \(f(x)\) then simplifies to \(\frac{5x^3}{x^2} - \frac{3x^2}{x^2} + \frac{10}{x^2}\), which simplifies further to \(5x - 3 + \frac{10}{x^2}\). Now, find the limit as \(x\) approaches infinity. The terms involving \(x\) in the denominator will approach 0, thus \(\lim _{x \rightarrow \infty} h(x) = 5x - 3\).

Find the limit of \(h(x) = \frac{f(x)}{x^{3}}\)

Next, divide the numerator of \(h(x)\) by \(x^{3}\). The function \(f(x)\) then simplifies to \(\frac{5x^3}{x^3} - \frac{3x^2}{x^3} + \frac{10}{x^3}\), which simplifies further to \(5 - \frac{3}{x} + \frac{10}{x^3}\). As before, find the limit as \(x\) approaches infinity. The terms with \(x\) in the denominator will approach 0, so \(\lim _{x \rightarrow \infty} h(x) = 5\).

Find the limit of \(h(x) = \frac{f(x)}{x^{4}}\)

Finally, divide the numerator of \(h(x)\) by \(x^{4}\). The function \(f(x)\) then simplifies to \(\frac{5x^3}{x^4} - \frac{3x^2}{x^4} + \frac{10}{x^4}\), which simplifies further to \(\frac{5}{x} - \frac{3}{x^2} + \frac{10}{x^4}\). When \(x\) approaches infinity, all the terms of the simplified \(h(x)\) will approach 0, leading to \(\lim _{x \rightarrow \infty} h(x) = 0\).

Key concepts

Calculus

Calculus is a branch of mathematics that deals with the study of change and motion. Its two main types are differential calculus, concerning rates of change and slopes of curves, and integral calculus, which deals with the accumulation of quantities and the areas under and between curves. In today’s exercise, we make use of differential calculus, specifically the concept of limits, to understand the behavior of functions as the input value grows indefinitely large, also known as limits at infinity.

Limits at infinity are crucial for analyzing function behavior for very large values of x, which is a common scenario in many areas of science and engineering. It helps us to predict the trend of a function's growth, whether it stabilizes to a particular value, grows without bound, or oscillates. By understanding these concepts, a student gains a toolkit for interpreting real-world problems mathematically.

Limit of a Function

The limit of a function is a fundamental concept in calculus that describes the behavior of a function as its argument approaches a particular value or infinity. In the context of the given exercise, we are investigating the limits of a function as the variable x heads towards infinity (\f\((x \rightarrow \f\)). Limits at infinity help us determine what value a function approaches as x becomes very large.

When we find \f\((\f_{x \rightarrow \f} h(x)\f\)), the goal is to see what the function 'settles into' as x grows. For example, direct substitution would typically lead to undefined expressions like 'infinity minus infinity' or 'infinity divided by infinity'. To tackle these instances, we often simplify the expressions by factoring, dividing, or other algebraic techniques to remove the indeterminate forms and find the limit value.

Polynomial Functions

Polynomial functions, such as \f\(f(x) = 5x^3 - 3x^2 + 10\f\), are expressions involving one or more algebraic terms, each consisting of a constant multiplied by a variable raised to a non-negative integer power. These types of functions are widely studied in algebra and calculus for their straightforward properties and predictable behavior.

In the exercise, we examine how different powers of x affect the polynomial when divided by \f\(x^2\f\), \f\(x^3\f\), and \f\(x^4\f\). As x goes to infinity, the influence of lower-power terms diminishes, and the highest-power term usually dominates. However, because we divide by a higher-power x term, the result can be quite different from the original polynomial's behavior at infinity. Hence, determining these limits is crucial for understanding long-term behavior of polynomials in relation to their degrees and leading coefficients.