Short on time? This page is the recap. For explanations, figures and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Notation: lim (x → a⁻) f(x) is the left-hand limit (x below a); lim (x → a⁺) f(x) is the right-hand limit (x above a); lim (x → a) f(x) is the two-sided limit.
Recap
- Follow the curve towards x = a from the left, then from the right. Each side gives a height, a y-value.
- Ignore x = a itself. Open circles, filled dots and gaps at a do not change the limit.
- The two-sided limit exists only when both one-sided limits exist and are equal.
- At an endpoint of the domain, only one one-sided limit can be read.
- A reading from a graph is an estimate, unless the graph labels exact coordinates.
Key relationships
| What you see at x = a | One-sided limits | Two-sided limit |
|---|---|---|
| Unbroken curve | Both equal f(a) | Exists, equals f(a) |
| Hole (open circle), maybe a separate dot | Both equal the height of the hole | Exists, equals the hole’s height, not the dot’s |
| Jump | Left ≠ right | Does not exist |
| Vertical asymptote | One or both are ±∞ (unbounded) | Does not exist |
| Faster and faster oscillation | Neither exists | Does not exist |
Writing ∞. lim (x → a) f(x) = ∞ is allowed when both sides grow without bound upwards. It explains why the limit does not exist. It is not a value.
Assumptions behind a graph reading
- The drawing is accurate at the scale shown.
- Nothing important happens between plotted points or outside the window.
- Open and filled circles are drawn where they belong. Most graphing tools draw neither, so holes can be invisible.
If you have the formula, check x-values where a denominator is 0 before trusting a picture.
Mistakes to avoid
- Giving f(a) (the filled dot) as the limit.
- Saying “undefined at a, so no limit”. A hole has a limit.
- Using only one side. At a jump, the right-hand limit may equal f(a), but the left side still disagrees.
- Reading x → a⁻ as “x is negative”. It means “from the left”.
- Calling ∞ a limit that exists.
- Giving the x-value instead of the height.
- Trusting a smooth-looking screen. A narrow asymptote can fall between plotted points.
Quick self-check
- A graph has an open circle at (4, 2) and a filled dot at (4, 7). What is lim (x → 4) f(x)? (2. The dot gives f(4) = 7, which does not affect the limit.)
- Find lim (x → 5⁻) 2|x − 5|/(x − 5) and lim (x → 5⁺) 2|x − 5|/(x − 5). Does the two-sided limit exist? (−2 and 2. No, the one-sided limits differ.)
- What can you say about lim (x → 1) 1/(x − 1)⁴? (Both sides grow without bound, so you may write = ∞. The limit does not exist as a real number.)
- Name the three graphical reasons a limit can fail to exist. (Left ≠ right, unbounded, oscillating.)
Next: practice questions.