Question

In Exercises 9 and \(10,\) find \(\lim _{x \rightarrow \infty} h(x),\) if possible. \(f(x)=5 x^{2}-3 x+7\) (a) \(h(x)=\frac{f(x)}{x}\) (b) \(h(x)=\frac{f(x)}{x^{2}}\) (c) \(h(x)=\frac{f(x)}{x^{3}}\)

Short answer

The limit as \( x \) approaches infinity of \( h(x) \) for the three cases are: (a) \( \infty \), (b) 5, and (c) 0.

Step-by-step solution

Simplifying the Function

First, simplify \( h(x) \) by dividing each term in \( f(x) \) by \( x \), \( x^{2} \), and \( x^{3} \) respectively: \n(a) \( h(x) = \frac{f(x)}{x} = \frac{5x^{2} - 3x + 7}{x} = 5x - 3 + \frac{7}{x} \)\n(b) \( h(x) = \frac{f(x)}{x^{2}} = \frac{5x^{2} - 3x + 7}{x^{2}} = 5 - \frac{3}{x} + \frac{7}{x^{2}} \)\n(c) \( h(x) = \frac{f(x)}{x^{3}} = \frac{5x^{2} - 3x + 7}{x^{3}} = \frac{5}{x} - \frac{3}{x^{2}} + \frac{7}{x^{3}} \)

Applying Limit Properties

Apply the limit properties to each simplified \( h(x) \):\n(a) \( \lim _{x \rightarrow \infty} h(x) = \lim _{x \rightarrow \infty} (5x - 3 + \frac{7}{x}) \)\n(b) \( \lim _{x \rightarrow \infty} h(x) = \lim _{x \rightarrow \infty}(5 - \frac{3}{x} + \frac{7}{x^{2}}) \)\n(c) \( \lim _{x \rightarrow \infty} h(x) = \lim _{x \rightarrow \infty}(\frac{5}{x} - \frac{3}{x^{2}} + \frac{7}{x^{3}}) \)

Calculate the Limits

The limit as \( x \) approaches infinity of a constant over \( x \) is \( 0 \). Thus, the limits are:\n(a) \( = 5\infty - 3 + 0 = \infty \) (the limit is positive infinity)\n(b) \( = 5 - 0 + 0 = 5 \) (the limit is 5)\n(c) \( = 0 - 0 + 0 = 0 \) (the limit is 0)

Key concepts

Calculus

Calculus is a branch of mathematics that deals with rates of change and the accumulation of quantities. At the heart of calculus are the concepts of differentiation and integration, which are used to analyze and solve problems involving change and area respectively. For students first learning calculus, it's essential to understand that it's not just about manipulating symbols. Instead, it's a tool for modeling the universe, to describe how things change and quantify the space around us. In the context of limits, calculus is concerned with understanding the behavior of functions as inputs approach certain values, including infinity. By mastering the principles of calculus, one can better comprehend natural phenomena and solve practical problems in a variety of fields, from physics to economics.

Infinite Limits

Infinite limits refer to the behavior of a function as it approaches either a specific value or infinity. When a limit approaches infinity, it means that the function grows without bound as the input grows larger. For example, as we see in our exercise with h(x) = 5x - 3 + \(\frac{7}{x}\), the term 5x grows indefinitely as x increases, leading to an infinite limit. Conversely, a function may also approach a finite limit or zero, even as the input grows large. This is exemplified in the exercises where h(x) = 5 - \(\frac{3}{x}\) + \(\frac{7}{x^{2}}\) results in a limit of 5 and h(x) = \(\frac{5}{x}\) - \(\frac{3}{x^{2}}\) + \(\frac{7}{x^{3}}\) results in a limit of zero. Understanding the concept of infinite limits is crucial for correctly evaluating the behavior of functions and grasping more complex calculus topics like improper integrals and asymptotic analysis.

Rational Functions

Rational functions are ratios of two polynomials. In the given exercise, we deal with functions like h(x) = \(\frac{f(x)}{x^n}\), where f(x) is a polynomial, and n is a positive integer. These functions often exhibit interesting behaviors as the variable approaches certain values, especially infinity. Simplifying rational functions by canceling common factors, as seen in the step-by-step solution, is a valuable technique. Doing so reveals how the function behaves for large values of x. In particular, if the degree of the numerator is less than or equal to the degree of the denominator, the function will have a horizontal asymptote, and the limit as x approaches infinity may be finite or zero. This core mathematic concept helps students analyze how complex relationships can be broken down into simpler parts for easier understanding and application in real-world scenarios.