A limit describes the value a function approaches as the input gets closer and closer to a chosen number. This idea matters because functions can have holes, jumps, or undefined points where direct substitution does not tell the full story. In calculus, limits let us study behavior near a point even when the function value at that point is missing or different.
They are the foundation for derivatives, integrals, continuity, and many approximation methods.
For a graph with a hole at x = a, the function may be undefined at x = a, but the y-values can still approach the same height L from both sides. In that case, the limit exists and is written lim x -> a f(x) = L. Direct evaluation asks for f(a), while a limit asks what f(x) approaches as x gets close to a but not necessarily equal to a.
Approximation uses nearby input values to estimate this approaching behavior, and the precise limit is the single value these estimates settle toward.
Key Facts
- lim x -> a f(x) = L means f(x) approaches L as x gets close to a.
- A limit can exist even if f(a) is undefined.
- Direct evaluation uses x = a, while a limit uses values of x near a.
- For a two-sided limit to exist, lim x -> a- f(x) = lim x -> a+ f(x).
- If the left-hand and right-hand limits are different, lim x -> a f(x) does not exist.
- A removable discontinuity is a hole where lim x -> a f(x) exists but f(a) is missing or not equal to the limit.
Vocabulary
- Limit
- A limit is the value a function approaches as the input gets arbitrarily close to a chosen number.
- Function value
- The function value f(a) is the output of the function when the input is exactly a.
- Left-hand limit
- A left-hand limit is the value f(x) approaches as x gets close to a from values less than a.
- Right-hand limit
- A right-hand limit is the value f(x) approaches as x gets close to a from values greater than a.
- Removable discontinuity
- A removable discontinuity is a hole in a graph where the limit exists but the function value is missing or different.
Common Mistakes to Avoid
- Using f(a) instead of the limit. This is wrong because f(a) describes the exact point, while the limit describes nearby behavior.
- Assuming a limit does not exist whenever the function is undefined. This is wrong because a limit can exist at a hole if both sides approach the same value.
- Checking only one side of the graph. This is wrong because a two-sided limit exists only when the left-hand and right-hand limits are equal.
- Thinking approximations prove the exact limit by themselves. This is wrong because numerical values suggest a limit, but the exact limit depends on the function's behavior as x gets arbitrarily close to a.
Practice Questions
- 1 For f(x) = (x^2 - 9)/(x - 3), find lim x -> 3 f(x) and state whether f(3) is defined.
- 2 A table gives f(1.9) = 4.81, f(1.99) = 4.9801, f(2.01) = 5.0201, and f(2.1) = 5.21. Estimate lim x -> 2 f(x).
- 3 A graph has an open circle at (4, 7) and a filled dot at (4, 2), with the curve approaching y = 7 from both sides. Explain the values of lim x -> 4 f(x) and f(4), and why they are not the same.