Math 115 Practice Midterm – Step-by-Step Calculus Guidance

Q10. Use the provided graph to find the following. If a limit or function value does not exist, write DNE and explain why it does not exist.

Background

Topic: Limits and Continuity

This question tests your understanding of how to evaluate limits and function values from a graph, as well as how to determine continuity at a point.

Key Terms and Concepts:

• Limit: The value that a function approaches as the input approaches a certain value.

• One-sided limits: The value the function approaches from the left ($x \to a^-$) or right ($x \to a^+$).

• Function value: The actual value of $f(x)$ at a specific $x$.

• Continuity: A function is continuous at $x=a$ if $\lim_{x \to a} f(x) = f(a)$ and the limit exists.

Step-by-Step Guidance

1. For each limit, identify the $x$-value in question and examine the graph as $x$ approaches that value from the left and right.

2. For one-sided limits (e.g., $\lim_{x \to 2^+} f(x)$), look only at the behavior as $x$ approaches from the specified side.

3. For $f(1)$, find the filled (solid) dot at $x=1$ on the graph. The $y$-value of this point is $f(1)$.

4. For continuity at $x=0$, check three things: (1) $f(0)$ exists, (2) $\lim_{x \to 0} f(x)$ exists, and (3) these two values are equal.

5. If a limit does not exist (DNE), look for jumps, holes, or asymptotes at that $x$-value and be ready to explain why.

Try solving on your own before revealing the answer!