A limit describes the value a function approaches as the input gets closer and closer to a chosen x-value. This idea matters because many functions behave in a predictable way near a point even if they are undefined or unusual exactly at that point. Numerical tables and graphs let you estimate this approaching value before using formal algebra.
Limits are the foundation for derivatives, integrals, continuity, and many models in science and engineering.
From a table, you inspect function values for x-values approaching the target from the left and from the right. From a graph, you trace the curve toward the target x-value and compare the y-values approached from both sides. If both sides approach the same y-value, the two-sided limit exists, even if the function value at the point is different or missing.
If the left and right behaviors disagree, grow without bound, or oscillate without settling, the limit does not exist.
Key Facts
- lim x->a f(x) = L means f(x) approaches L as x gets close to a.
- The left-hand limit is written lim x->a- f(x), using x-values less than a.
- The right-hand limit is written lim x->a+ f(x), using x-values greater than a.
- lim x->a f(x) exists only if lim x->a- f(x) = lim x->a+ f(x).
- The value f(a) can be different from lim x->a f(x), or f(a) may be undefined.
- For a removable hole, simplify the expression if possible, then evaluate the simplified form at x = a to find the limit.
Vocabulary
- Limit
- A limit is the value a function approaches as the input approaches a specific number.
- Left-hand limit
- A left-hand limit is the value approached by a function as x gets closer to a target value from smaller x-values.
- Right-hand limit
- A right-hand limit is the value approached by a function as x gets closer to a target value from larger x-values.
- Two-sided limit
- A two-sided limit exists when the left-hand and right-hand limits both approach the same value.
- 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) as the limit automatically is wrong because a limit depends on nearby values, not just the value at the target point.
- Checking only one side of the graph is wrong because a two-sided limit exists only when both sides approach the same y-value.
- Reading an open circle as the function value is wrong because an open circle usually marks an approached value that is not actually included at that x.
- Assuming a limit does not exist whenever f(a) is undefined is wrong because the surrounding function values may still approach one common value.
Practice Questions
- 1 A table for f(x) near x = 2 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).
- 2 For g(x) = (x^2 - 9)/(x - 3), use algebra and nearby values to find lim x->3 g(x).
- 3 A graph has an open circle at (1, 4), a filled dot at (1, 2), and the curve approaches y = 4 from both the left and right. Explain the value of lim x->1 f(x) and how it differs from f(1).