Guided Practice: Evaluating Limits Using Graphs

Q2. Use the graphs of f and g to evaluate the following limits:

1. limx→2−(f(x) − g(x))

2. limx→2+(f(x) − g(x))

3. limx→2(f(x) − g(x))

4. limx→2+ \( \frac{f(x)}{g(x)} \)

5. f(2) + g(2) + limx→−1g(x)

6. limx→4(x2g(x))

7. f(0) + limx→4f(x)

8. limx→4 3(f(x) + g(t))

Background

Topic: Limits and Graphical Analysis

These questions test your ability to evaluate limits and function values using the graphs of two functions, f(x) and g(x). You will need to interpret left-hand and right-hand limits, as well as function values at specific points, by carefully reading the graphs provided.

Key Terms and Concepts:

• Left-hand limit (limx→a− f(x)): The value that f(x) approaches as x approaches a from the left.

• Right-hand limit (limx→a+ f(x)): The value that f(x) approaches as x approaches a from the right.

• Limit at a point (limx→a f(x)): Exists if and only if both the left-hand and right-hand limits exist and are equal.

• Function value (f(a)): The actual value of the function at x = a, which may or may not be the same as the limit.

• Difference of functions: (f(x) - g(x)) means you subtract the y-values of g(x) from f(x) at the same x.

• Product and quotient of functions: Multiply or divide the y-values as indicated.

Step-by-Step Guidance

1. For each limit, first identify the relevant x-value (e.g., x = 2, x = -1, x = 4, etc.).

2. For left-hand limits (x→a−), trace the graph as x approaches a from values less than a. For right-hand limits (x→a+), trace from values greater than a.

3. Read the y-values of f(x) and g(x) from the graphs at the specified points or as x approaches those points. Pay attention to open and closed circles, which indicate whether the function is defined at that point.

4. For expressions like f(x) - g(x), subtract the y-values you found for f(x) and g(x) at the relevant limit.

5. For expressions involving products or quotients, multiply or divide the y-values as appropriate, being careful with undefined values (e.g., division by zero).

Try solving on your own before revealing the answer!