Problem 110 - Algebra and Trigonometry

Question

Use everyday language to describe the behavior of a graph near its vertical asymptote if \(f(x) \rightarrow \infty\) as \(x \rightarrow-2^{-}\) and \(f(x) \rightarrow-\infty\) as \(x \rightarrow-2^{+}\).

Step-by-Step Solution

Answer

As \(x\) approaches -2 from the left, \(f(x)\) increases without limit, and as \(x\) approaches -2 from the right, \(f(x)\) decreases without limit. This indicates a vertical asymptote at \(x = -2\). In everyday language, the function goes upwards forever when you approach -2 from the left, and downwards forever when you approach from the right.

Step 1: Understand the limit

First, recognize what the given information is describing. The function \(f(x)\) behaves differently as \(x\) approaches -2 from the left (\(x \rightarrow -2^{-}\)) and the right (\(x \rightarrow -2^{+}\)). This means that there is a vertical asymptote at \(x = -2\).

Step 2: Describe the behavior as \(x \rightarrow -2^{-}\)

When \(x\) approaches -2 from the left, \(f(x)\) goes towards positive infinity. This means that as \(x\) gets very close to -2 from the left side, the values of the function get larger and larger in the positive direction. So, if you were to visually represent the function, the graph would be going upwards without end as it gets closer to \(x = -2\).

Step 3: Describe the behavior as \(x \rightarrow -2^{+}\)

When \(x\) approaches -2 from the right, \(f(x)\) goes towards negative infinity. This means that as \(x\) gets very close to -2 from the right side, the values of the function get larger and larger in the negative direction. So, visually, the graph would be going downwards without end as it gets closer to \(x = -2\).