Question

In Exercises 13-20, use graphs and tables to find the limits. $$\lim _{x \rightarrow-3^{+}} \frac{x}{x+3}$$

Short answer

The limit of the function \(\frac{x}{x+3}\) as \(x\) approaches -3 from the right is -∞.

Step-by-step solution

Understand the function

Analyze the function \(\frac{x}{x+3}\). It's a rational function, so it could potentially have a vertical asymptote, which we'll need to investigate. From the form of the function, there might be an asymptote at \(x = -3\), which means the function behaves in an extreme way at \(x = -3\).

Graph the function

Sketch the function \(\frac{x}{x+3}\) or generate its graph via graphing calculator or online software. Pay attention to the behavior of the function as \(x\) approaches -3.

Create a table

A table with values of \(x\) approaching -3 from the right side can be created. As we're interested in \(x \rightarrow -3^{+}\), our \(x\)-values can be -2.9, -2.99, -2.999, etc. Calculate corresponding \(y\)-values for these \(x\)-values.

Analyze the table and graph

As \(x\) approaches -3 from the right, we observe that the function value approaches -∞. This indicates the existence of a vertical asymptote at \(x = -3\). Since the question asks for \(x \rightarrow -3^{+}\), we can conclude from the graph and table that the limit is negative infinity.