Problem 78 - College Algebra

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

The graph of the function \(f(x)\) surges upward towards positive infinity as \(x\) approaches -2 from the left, and plunges downward towards negative infinity as \(x\) approaches -2 from the right, creating a stark 'split' or 'divide' in the graph at \(x = -2\) and forming a vertical asymptote.

Step 1: Understand the function behavior on the left of the asymptote

On observing, we can see that as the value of \(x\) comes close to -2 from the left side (e.g., -2.1, -2.01, -2.001, etc.), the value of \(f(x)\) increases significantly towards positive infinity. This behavior can be seen as climbing a mountain with its peak pointing towards the sky (positive infinity).

Step 2: Understand the function behavior on the right of the asymptote

Simultaneously, as \(x\) approaches -2 from the right side (e.g., -1.9, -1.99, -1.999, etc.), the value of \(f(x)\) decreases sharply towards negative infinity. It's like digging a hole that gets deeper and deeper towards the bottom (negative infinity).

Step 3: Visualizing a vertical asymptote

When seen collectively, the function near the asymptote at \(x = -2\), from left it shoots upwards towards the sky while from right it plunges downwards towards the bottom creating a gap that cannot be reached or crossed, mounting a vertical wall at \(x = -2\). This vertical wall is the vertical asymptote.