How to read limits and function values from a graph

Key concept: When a problem points to a graph, the picture is the data. You are not guessing formulas; you are reading what the graph actually does near the target x-value.

Step by step

To evaluate expressions from a graph, use these rules:

• $\lim_{x\to a^-} f(x)$: follow the graph from the left toward $x=a$
• $\lim_{x\to a^+} f(x)$: follow the graph from the right toward $x=a$
• $\lim_{x\to a} f(x)$: check whether left and right limits match
• $f(a)$: look for the filled-in point at $x=a$

For a problem like this, I would read each requested value directly from the graph and compare the left and right behavior at the same $x$-coordinate.

Since no graph is included here, the exact numerical answers cannot be determined from the text alone. If you have the graph, look carefully at:

1. open circles vs. filled dots
2. whether the curve approaches the same $y$-value from both sides
3. any jump, hole, or vertical asymptote near the target point

If you want, I can also turn the graph into a table of answers once the image is available.

Pitfall alert

People often mix up $f(a)$ with $\lim_{x\to a} f(x)$. A filled dot tells you the function value; the approach from either side tells you the limit. Another issue is reading the wrong point because of small scale or unlabeled axes.

Try different conditions

If the graph has a hole at $x=2$ but both sides approach the same height, then $\lim_{x\to 2} f(x)$ exists even though $f(2)$ may be undefined. If the graph jumps at $x=2$, then the one-sided limits differ and the two-sided limit does not exist.

Further reading

graph of a function, function value, two-sided limit