AP® Calculus AB · Unit 1 · Topic 1.2
One-Sided Limits and When a Limit Fails to Exist
A one-sided limit looks at a function's height from just one direction. The two-sided limit exists only when the left and right limits agree on the same value.
A limit asks where a function is heading as x closes in on some point, but there are two roads into that point: from the left and from the right. The left-hand limit, written lim as x → c^-, tracks the heights as x creeps up toward c from values smaller than c. The right-hand limit, lim as x → c^+, tracks the heights as x creeps down toward c from values bigger than c. Nothing forces these two approaches to land in the same place. A graph can climb toward one height from the left and a completely different height from the right.
The two-sided limit, plain lim as x → c, only exists when both one-sided limits exist and equal the same number. If they disagree, the two-sided limit fails to exist, even if the function is perfectly defined at c. That's the key separation: existence of f(c) and existence of the limit are different questions. A jump discontinuity is exactly this situation: the function has a real, solid value at c, but the left and right limits point to different heights, so there is no single number the function is 'heading toward.'
This idea sets up the formal definition of continuity later in the unit. Continuity demands three things line up: f(c) exists, the two-sided limit exists, and they're equal. One-sided limits are also how you handle piecewise functions and absolute-value expressions, where the formula itself changes depending on which side of a point you're standing on. Oscillating functions push the idea further: near a wild oscillation, the one-sided limits themselves can fail to settle on any single value, so the limit fails for a different reason than a jump.
Key terms
- left-hand limit
- The value a function approaches as x gets close to a point from values smaller than it.
- right-hand limit
- The value a function approaches as x gets close to a point from values larger than it.
- two-sided limit
- The single value a function approaches that exists only when the left-hand and right-hand limits agree.
- jump discontinuity
- A break in a graph where the left and right limits are both finite but unequal.
Watch it explained
- How it worksHow do the left-hand and right-hand limits combine into a single two-sided limit?
- Worked exampleFind the left and right limits of |x|/x as x approaches 0 and say why the limit fails.
- Common mistakeWhy isn't a jump in the graph the same thing as the function being undefined?
- Going furtherWhat happens to the limit when a function oscillates faster and faster near zero?
Check yourself
End of topic.