Calculus How To

Determining Limits From a Graph

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Problem Solving > Determining Limits From a Graph

You can’t get an exact figure for a limit from a graph, but you can get a very good approximation. When determining limits from a graph, look for y-values (called function values) near the x-value in the question.

Determining Limits From a Graph: Examples

The following examples are based on this graph of a piecewise function which has a jump at x = 1:
determining limits from a graph

Example question 1: What is the limit of f(x) as x approaches 2?

Solution: “f(x)” is the function value at 2 (a.k.a. the y-value). We want to know what’s happening to the y-values around x = 2. Sketch a couple of arrows going to x = 2 from both sides and the answer should be clear:

The green arrows show that as we approach x = 2 from both ends of the function, the y-value gets closer and closer to zero. Our answer: f(x) = 0.

Note: We never say “exactly” at 2, because we can’t be sure from a rudimentary graph like this that there’s not anything strange happening, like a pinprick hole. You can usually only find strange behavior like this algebraically, not graphically.

Example question 2: What is the limit of f(x) as x approaches 4?

Solution:This time, we have a problem, which becomes clear once we draw our green arrows:

The top arrow is pointing towards a y-value of 4. But the bottom arrow isn’t even on the line of the function…it’s sitting somewhere out in space. In order to determine a limit from a graph:

  • Both green arrows must point to the same number and
  • Both arrows must be on the function’s line.

Both arrows are not on the function’s line, so the limit does not exist.

Example question 3: What is the limit as x approaches 1?


Both arrows are not pointing to the same number, so the limit does not exist.

However, there are a couple of special limits that are denoted with a plus or minus sign (lim+ or lim-):

  • The left-hand arrow is approaching y = -1, so we can say that the limit from the left (lim →-) is f(x) = -1.
  • The right hand arrow is pointing to y = 2, so the limit from the right (lim →+) also exists and is f(x) = 2.
Stephanie Glen. "Determining Limits From a Graph" From Calculus for the rest of us!

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