Limits from a Graph – Step-by-Step Guidance

Q1. Use the graph of $f(x)$ to find the following limits and function values:

• $\lim\limits_{x \to -6^-} f(x)$

• $\lim\limits_{x \to -6^+} f(x)$

• $\lim\limits_{x \to -6} f(x)$

• $f(-6)$

• $\lim\limits_{x \to -3^-} f(x)$

• $\lim\limits_{x \to -3^+} f(x)$

• $\lim\limits_{x \to -3} f(x)$

• $f(-3)$

• $\lim\limits_{x \to 3^-} f(x)$

• $\lim\limits_{x \to 3^+} f(x)$

• $\lim\limits_{x \to 3} f(x)$

• $f(3)$

• $\lim\limits_{x \to 6^-} f(x)$

• $\lim\limits_{x \to 6^+} f(x)$

• $\lim\limits_{x \to 6} f(x)$

• $f(6)$

Background

Topic: Limits from Graphs

This question tests your ability to evaluate one-sided and two-sided limits, as well as function values, by interpreting the graph of a function. You will need to distinguish between left-hand limits, right-hand limits, and the actual value of the function at a point.

Key Terms and Concepts:

• Left-hand limit: $\lim\limits_{x \to a^-} f(x)$ is the value $f(x)$ approaches as $x$ approaches $a$ from the left.

• Right-hand limit: $\lim\limits_{x \to a^+} f(x)$ is the value $f(x)$ approaches as $x$ approaches $a$ from the right.

• Two-sided limit: $\lim\limits_{x \to a} f(x)$ exists if and only if both one-sided limits exist and are equal.

• Function value: $f(a)$ is the actual value of the function at $x = a$ (may be different from the limit).

Step-by-Step Guidance

1. For each limit, locate the $x$-value of interest on the $x$-axis of the graph (e.g., $x = -6$, $x = -3$, $x = 3$, $x = 6$).

2. To find a left-hand limit ($x \to a^-$), trace the graph as $x$ approaches $a$ from values less than $a$ (from the left). Observe the $y$-value the graph approaches.

3. To find a right-hand limit ($x \to a^+$), trace the graph as $x$ approaches $a$ from values greater than $a$ (from the right). Observe the $y$-value the graph approaches.

4. For the two-sided limit ($x \to a$), check if the left-hand and right-hand limits are equal. If they are, that is the limit. If not, the two-sided limit does not exist.

5. To find the function value $f(a)$, look for a solid dot at $x = a$ on the graph. If there is a solid dot, its $y$-value is $f(a)$. If there is an open circle, $f(a)$ is not that value.

Try solving on your own before revealing the answer!