Question

In Exercises \(1-8,\) use graphs and tables to find (a) \(\lim _{x \rightarrow \infty} f(x)\) and (b) \(\lim _{x \rightarrow-\infty} f(x)\) (c) Identify all horizontal asymptotes. $$f(x)=\frac{3 x^{3}-x+1}{x+3}$$

Short answer

The limit of the function as \(x \rightarrow \infty\) is infinite. The limit of the function as \(x \rightarrow -\infty\) is infinite. The function has no horizontal asymptotes.

Step-by-step solution

Finding the Limit as \(x\) approaches infinity

To find the limit of the function as \(x \rightarrow \infty\), we use polynomial division to divide the numerator and the denominator. On doing the division, we consider only the highest degree term (ignoring the remaining terms), we get \(3x^2\). So \(\lim_{{x \rightarrow \infty}} f(x) = \lim_{{x \rightarrow \infty}} 3x^2 = \infty\). Thus, the limit as \(x\) approaches infinity does not exist since it tends to infinity.

Finding the Limit as \(x\) approaches negative infinity

Similarly, to find the limit of the function as \(x \rightarrow -\infty\), we consider only the highest degree term after the polynomial division i.e \(3x^2\). So \(\lim_{{x \rightarrow -\infty}} f(x) = \lim_{{x \rightarrow -\infty}} 3x^2 = \infty\). Thus, the limit as \(x\) approaches negative infinity does not exist since it tends to infinity.

Identifying Horizontal Asymptotes

A horizontal asymptote of a function is a horizontal line \(y = k\) where the value of the function \(f(x)\) approaches \(k\) as \(x\) goes to infinity or negative infinity. In this case, as \(x\) goes to infinity or negative infinity, the value of the function does not approach a finite number, rather it approaches infinity. Therefore, there are no horizontal asymptotes for this function.

Key concepts

Horizontal Asymptotes

Understanding horizontal asymptotes is crucial when analyzing the behavior of functions as they go towards infinity. A horizontal asymptote represents the value that a function's output will approach as the input either increases or decreases without bound. In more formal terms, if the limit of a function as the variable approaches positive or negative infinity equals some constant value, then this constant is a horizontal asymptote.

For the function given in the exercise, \(f(x)=\frac{3x^3-x+1}{x+3}\), we analyze its limit as \(x\) approaches infinity (\(\infty\)) and negative infinity (\(\-\infty\)). However, as indicated in the solution, the limit of this specific function as \(x\) increases or decreases without bound is infinity, not a finite number. This implies that there aren't any horizontal asymptotes for this function. It's key to note, for a rational function to have a horizontal asymptote, the degree of the polynomial in the numerator must be less than or equal or the same to that in the denominator. If the degree is higher, as in this case, the function will have no horizontal asymptotes because the function's growth outpaces the value that a horizontal asymptote would represent.

Polynomial Division

Polynomial division, particularly when applied within the context of limits, helps simplify function expressions to more easily find these limits at infinity. The core idea is to divide the terms of highest degree in the numerator by the terms of highest degree in the denominator so that you can evaluate the behavior of the function as the variable approaches infinity.

In the solution provided, polynomial division was used to simplify \(f(x)=\frac{3x^3-x+1}{x+3}\) to \(3x^2\) by considering only the highest power terms of \(x\). This simplification underscores that, as \(x\) goes to either positive or negative infinity, the lower degree terms become negligible in comparison to the dominance of the highest degree terms. Thus, we find the limits are infinite, making the process of finding horizontal asymptotes unnecessary in this scenario. It's important to understand that polynomial division is not just a mechanical process but a way to distill the essence of a function's long-term behavior.

Graphical Analysis of Limits

A graphical analysis provides a visual representation of a function and offers insight into its end-behavior—how the function acts as \(x\) approaches infinity or negative infinity. It's an excellent tool for visualizing limits and asymptotic behavior. For the exercise's function \(f(x)\), plotting the graph would show how the function behaves without simplifying it algebraically.

While the algebraic approach has determined that there are no horizontal asymptotes, graphing the function would nonetheless show \(f(x)\) becoming steeper and tending toward infinity as \(x\) grows large in either the positive or negative direction. Such a graphic illustration is helpful as it visually confirms the function's lack of horizontal asymptotes and complements the algebraic findings. Students should understand that graphing is not just a means to an end but also a powerful explanatory tool, which can help solidify the concept of limits, particularly at infinity.