Guided Practice: Infinite Limits and Limits at Infinity Using Graphs

Q1. Use the graph of $f(x)$ shown to find the indicated limits.

Background

Topic: Infinite Limits and Limits at Infinity

This question tests your understanding of how to evaluate limits using the graph of a function, especially when the limits approach infinity or negative infinity, or when the function approaches a vertical or horizontal asymptote.

Key Terms and Formulas

• Limit: The value that $f(x)$ approaches as $x$ approaches a certain value.

• Infinite Limit: When $f(x)$ increases or decreases without bound as $x$ approaches a certain value.

• Vertical Asymptote: A line $x = a$ where $f(x)$ increases or decreases without bound as $x$ approaches $a$.

• Horizontal Asymptote: A line $y = L$ where $f(x)$ approaches $L$ as $x$ approaches infinity or negative infinity.

Step-by-Step Guidance

1. Carefully examine the graph of $f(x)$ provided. Identify any vertical or horizontal asymptotes, as these often indicate where limits may approach infinity or a constant value.

   

2. For each limit, determine the direction from which $x$ is approaching (from the left, right, or both sides) and observe the behavior of $f(x)$ as $x$ gets close to the specified value.

3. For limits as $x$ approaches a finite value (e.g., $x \to 3$ or $x \to -2$), check if the function goes to infinity, negative infinity, or a finite value. If the function goes up or down without bound, the limit is infinite (positive or negative).

4. For limits as $x$ approaches infinity or negative infinity, look for horizontal asymptotes or the end behavior of the graph. What value does $f(x)$ approach as $x$ gets very large or very small?

5. Write the limit notation for each part and use the graph to estimate the value or determine if the limit is infinite.

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