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Chapter 3: Problem 110

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 \(x \rightarrow-2^{+}\)

Short Answer

Expert verified
At x = -2, there is a vertical asymptote. As the graph nears -2 from the left side, it shoots upward towards infinity. As the graph nears -2 from the right side, it plunges downward into negative infinity.

Step by step solution

01

Understanding the concept of vertical asymptotes

A vertical asymptote represents a value \(x = a\) that the function approaches but never quite reaches. A function behaves differently when approaching a vertical asymptote from either side of the number line. It will either steadily increase or decrease without taking on the value of the asymptote.
02

Understanding the behavior of the function as x approaches -2 from the left

If \(f(x) \rightarrow \infty\) as \(x \rightarrow -2^{-}\), it means that as x values get closer and closer to -2 from the left side (or lesser side), the y-values of the function climb higher and higher. The function \(f(x)\) tends to positive infinity. This implies the graph shoots upward as it gets close to -2 from the left.
03

Understanding the behavior of the function as x approaches -2 from the right

If \(f(x) \rightarrow -\infty\) as \(x \rightarrow -2^{+}\), it means that as x values get closer and closer to -2 from the right side (or greater side), the y-values of the function plummet down or become very large negative values. The function \(f(x)\) tends to negative infinity. This implies as the graph moves towards -2 from the right, it plunges downward into negative infinity.
04

Analyzing and describing the overall behaviour of the graph near its vertical asymptote

Having understood the trends on both sides, it can be concluded that at x = -2, the graph has a vertical asymptote. As we approach this vertical asymptote from the left, the graph shoots upward and as we approach it from the right, the graph plunges downward.

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