Math Problem Statement

Use the graph of the rational function to complete the following statement: As x approaches -3 from the left, f(x) approaches ____.

Solution

Looking at the graph of the rational function, we are interested in determining the behavior of the function f(x)f(x) as xx approaches 3-3 from the left (x3x \to -3^-).

From the graph, it appears that as xx approaches 3-3 from the left, the value of f(x)f(x) decreases without bound. This indicates that f(x)f(x) \to -\infty as x3x \to -3^-.

Thus, the correct statement is:

As x3, f(x).\text{As } x \to -3^-, \ f(x) \to -\infty.

Would you like further details or explanation on this?

Here are five questions that could expand on this topic:

  1. What is the behavior of f(x)f(x) as x3+x \to -3^+?
  2. Can we describe the behavior of f(x)f(x) at other asymptotes?
  3. How do vertical asymptotes influence the behavior of rational functions?
  4. How would the graph behave if there were a horizontal asymptote?
  5. What changes if we approach a different vertical asymptote, like x=3x = 3?

Tip: Always observe vertical asymptotes to understand the behavior of a function as xx approaches certain critical values.

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Math Problem Analysis

Mathematical Concepts

Rational Functions
Asymptotic Behavior
Limits

Formulas

Limit of a function as x approaches a vertical asymptote

Theorems

Behavior of Rational Functions near Vertical Asymptotes

Suitable Grade Level

Grades 10-12