Short on time? This page is the recap. For explanations, the graph and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Notation: lim (x → a) f(x) means “the limit as x approaches a of f(x)”; a⁻ and a⁺ mean from the left and from the right.
Recap
- A limit is the value f(x) approaches as x gets close to c, not the value f(c).
- The two-sided limit exists only if both one-sided limits exist and are equal.
- A limit fails to exist when the sides disagree, when the function is unbounded, or when it oscillates.
- Tables and graphs give estimates. A table can skip behaviour between its inputs; a graph can hide a hole at the wrong scale.
Key relationships
| Representation | Read the limit by… | Trap |
|---|---|---|
| Graph | Covering up x = c and following the curve from each side | Reading the filled dot (that is f(c)) |
| Table | Following the trend of outputs from each side | Copying the nearest entry; trusting too few inputs |
| Piecewise formula | Using the piece that applies on each side | Using the rule for x = c itself |
| Words | “Approaches … from the right” → lim (x → c⁺) | Treating “approaches” as “equals at” |
| Sum, difference or product | Combining the limits, if each exists | Using the property when one limit does not exist: split into sides instead |
| Quotient | Dividing the limits, if each exists and the bottom limit is not 0 | Dividing by a limit of 0 |
| Composite f(g(x)) | Finding L = lim g(x) and the side; then the matching limit of f at L | Using f(L), or ignoring the side |
Assumptions behind the method
- A table is only useful if you may assume its trend continues near c. Exam questions usually say so.
- The sum, difference, product and quotient properties need each separate limit to exist (and a nonzero bottom limit for a quotient).
- For a composite, the side matters whenever f has different one-sided limits at L. If g(x) can equal L near c, the value f(L) matters too.
Mistakes to avoid
- Reading f(c) from a filled dot and calling it the limit.
- Saying “no limit” too early. (x + 1) · f(x) can have a limit even if f does not.
- Ignoring the side in a composite limit.
- Using f(L) instead of the limit of f at L when f has a hole at L.
- Testing one side only in a table, then claiming a two-sided limit.
- Trusting a table of special inputs, such as sin(π/x) at x = 0.1, 0.01, 0.001.
Quick self-check
- f(x) = 4 − x for x < 1, f(x) = x² + 2 for x > 1, and f(1) = 0. Find lim (x → 1) f(x). (3: both sides give 3; f(1) = 0 is irrelevant.)
- lim (u → 0⁻) f(u) = 5 and lim (u → 0⁺) f(u) = −2. The values of g(x) approach 0 from above as x → 4. Find lim (x → 4) f(g(x)). (−2: the input to f arrives from the right.)
- Write in notation: “as t gets close to 6 from the right, D(t) gets close to 0.5”. (lim (t → 6⁺) D(t) = 0.5)
Next: practice questions.