This Calculus Problem of the Week (PoW) focuses on determining limits from a graph. These questions are meant as a quick check of your understanding, and mimic the type of typical question you’ll find on a college-level or AP®-style exam.

As multiple-choice questions you get immediate quick feedback, and then more detailed solutions and discussion are further down the screen. While most students quickly learn how to answer these questions correctly (which is great!), we’d like to note that understanding what a limit actually is requires more work. After all, the actual definition of “limit” that underlies all(!) of Calculus took centuries to develop – really! The interactive materials in our Limits Chapter are designed to help you do just that, and it’s all free, because we believe every dedicated student deserves access to high-quality materials to use so they (you) can learn well and excel in this and future classes.

Here’s this week’s problem:

Practice Problem: Limits from a Graph

Use the graph of $y = f(x)$ to answer the following questions.

Solution details are below.

Solution details are below.

Solution details are below.

Solution details are below.

Solution details are below.

More detailed solutions.

Difference between function’s output value and the limit value

Questions (a) and (b) focus on the difference between the value of the limit at a given input value, and the function’s output value at that same input value. If you’d like to focus more on this distinction, please visit our Limits Introduction topic. We also address there some common (and completely understandable) misconceptions beginning students have about limits that cause unnecessary confusion later.

Each of our answers is based on the working definition of “limit” we use early in the course, before we’ve developed a more formal definition:

The limit of the function $f(x)$ as x approaches a, written \[\lim_{x \to a} f(x) = L\]is a number L (if one exists) such that $f(x)$ is as close to L as we want
whenever x is sufficiently close to a.

(a) As we approach $x = 0$ from either side, the output value we can get as close to as we’d like is \[\lim_{x \to 0}f(x) = 2.5 \quad \cmark\] even though the function never attains that value. This is illustrated by the red arrows pointing toward the point $(0, 2.5)$ on the graph below.

(b) The function’s output value at $x = 0$ is \[f(0) = 1 \quad \cmark\] as illustrated by the red star surrounding the solid dot at the point $(0, 1)$ on the graph above.

One-sided limits

Please visit One-sided Limits for a discussion of this topic and more practice problems. The key things to remember:

Otherwise the limit does not exist (DNE).

(c) As we approach $x = 1.5$ from the left, the output value we can get as close to as we’d like is \[\lim_{x \to 1.5^-} = 2 \quad \cmark \] as illustrated by the arrow pointing from the left toward the open point $(1.5, 2)$ on the graph below.

(d) As we approach $x = 1.5$ from the right, the output value we can get as close to as we’d like is \[\lim_{x \to 1.5^+} = -2.5 \quad \cmark \] as illustrated by the arrow pointing from the right toward the open point $(1.5, -2.5)$ on the graph above.

(e) As we saw in parts (c) and (d), \[\lim_{x \to 1.5^-}f(x) \ne \lim_{x \to 1.5^+}f(x)\] and hence

$\displaystyle{\lim_{x \to 1.5}f(x)}$ does not exist (DNE) $\quad \cmark$

What questions or thoughts do you have about any of the above? Please Comment below, or post on our Forum. Again, it’s all free for your use: our only aim is to help you do well in your class!

And if we’ve helped you learn something, or verify your understanding, please consider giving a little something back and

In next week’s problem, you can check your ability to compute typical limits that you absolutely must be able to find, and that we guarantee will be on an upcoming exam.