A limit describes the value a function approaches as x gets close to a chosen input. Sometimes there is no single value that the function settles toward, so the limit does not exist. This matters because limits are the foundation of continuity, derivatives, and integrals.
Recognizing failure cases helps students interpret graphs and avoid forcing an answer where none exists.
A two-sided limit exists only when the left-hand and right-hand limits exist and are equal. Limits commonly fail when the two sides approach different values, when the function oscillates without settling, or when values grow without bound. A hole or a jump in the graph does not automatically mean the limit fails, so the behavior near the point is more important than the function value at the point.
Graphs, tables, and formulas all must be checked for the same idea: approach, not arrival.
Key Facts
- lim x -> a f(x) exists if and only if lim x -> a- f(x) = lim x -> a+ f(x) = L.
- If lim x -> a- f(x) and lim x -> a+ f(x) are different, then lim x -> a f(x) does not exist.
- If f(x) grows without bound as x approaches a, then the finite limit does not exist.
- For f(x) = 1/(x - a)^2, lim x -> a f(x) = infinity, so there is no finite limit.
- For f(x) = 1/(x - a), lim x -> a- f(x) = -infinity and lim x -> a+ f(x) = infinity, so the two-sided limit does not exist.
- For f(x) = sin(1/x), lim x -> 0 f(x) does not exist because the function oscillates between -1 and 1 infinitely often.
Vocabulary
- Limit
- A limit is the value a function approaches as the input gets closer to a specified number.
- One-sided limit
- A one-sided limit describes what a function approaches from only the left or only the right side of an input.
- Two-sided limit
- A two-sided limit describes what a function approaches when the input gets close from both sides.
- Oscillation
- Oscillation occurs when a function keeps moving between different values instead of approaching one value.
- Unbounded growth
- Unbounded growth occurs when function values increase or decrease without limit as the input approaches a point.
Common Mistakes to Avoid
- Assuming a limit does not exist just because f(a) is undefined. This is wrong because a limit depends on nearby values, not on the value at x = a.
- Ignoring one-sided limits. This is wrong because a two-sided limit exists only when the left-hand and right-hand limits both exist and match.
- Writing infinity as the limit value without context. This is wrong because infinity is not a real number, so the finite limit does not exist even if the function grows without bound.
- Using a table with too few values to prove a limit exists. This is wrong because a table can suggest behavior, but jumps or oscillations may be missed without graph or algebraic analysis.
Practice Questions
- 1 For f(x) = (x^2 - 4)/(x - 2), find lim x -> 2 f(x), or state that it does not exist.
- 2 For f(x) = 1/(x - 3), find lim x -> 3- f(x), lim x -> 3+ f(x), and decide whether lim x -> 3 f(x) exists.
- 3 A graph approaches y = 2 from the left of x = 1 and y = 5 from the right of x = 1. Explain whether lim x -> 1 f(x) exists and why.