Precalculus evaluating limits from a graph worksheet
In Precalculus reading limits off a graph is where limit notation is learned: one-sided limits, limits that are infinite at a vertical asymptote, and limits at infinity that describe end behavior and horizontal asymptotes.
Also taught in Calculus, at its own level and with its own examples.
Select your difficulty
Pick more than one for a sheet that mixes them.
The preview can look cramped on a phone. The PDF and printed sheet come out normal.
A graph has a vertical asymptote at x=3, falling on the left and rising on the right. Find the limits at 3
From the left the curve falls without bound: the left limit is negative infinity.
From the right it rises without bound: the right limit is infinity.
The two sides disagree, so the two-sided limit does not exist.
The answer is x→3−limf(x)=−∞, x→3+limf(x)=∞, two-sided DNE.
Where students go wrong
Mixing up the side the arrow points to. The minus in 3⁻ means coming from values less than 3, which is the left side of the asymptote.
Also called limits graphically, limits from a graph, one-sided limits from a graph, reading limits off a graph, jump discontinuity, removable discontinuity, vertical asymptote or limits at infinity from a graph.
What you can put on this worksheet
Limits at a break in the curve
Use the graph to find x→1−limf(x)→ 3
Limits at a vertical asymptote
Use the graph to find x→2−limf(x)→ −∞
Limits at infinity
Use the graph to find x→∞limf(x)→ 0
Questions about these worksheets
Yes. Every evaluating limits from a graph sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.
Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.
Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.
Precalculus is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.
Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.
That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the evaluating limits from a graph questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.
Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.
You pick from 3: limits at a break in the curve, limits at a vertical asymptote and limits at infinity. Tick as many as you want and set how many of each, or let it spread them evenly.
Reading the filled dot instead of the height the curve heads for. Where the curve approaches 3 but the solid dot sits at 1, the limit is 3 and the value of the function is 1, and a question asking for the limit does not care where the dot is.