Nearby input
The expression x → c describes inputs approaching c, not necessarily taking the value c.
Approach · one-sided behavior · continuity · precision
Limits & Continuity
Limits describe what function values approach as inputs move toward a target. They let calculus reason about local behavior even when the function value at the target is missing, different, or irrelevant, and they provide the limiting definitions behind derivatives and integrals.
What the notation separates
Approaching a point and evaluating at the point are different operations.
That separation is what makes holes, asymptotes, endpoint behavior, derivative definitions, and many piecewise constructions mathematically manageable.
One-sided behavior
A two-sided finite limit requires the left-hand and right-hand limits to exist and agree.
Function value
The value f(c) can differ from the limit or fail to exist entirely; the limit is controlled by nearby behavior.
Continuity
Continuity at c adds the requirement that f(c) exists and equals the two-sided limit as x approaches c.
Lessons · registry-backed navigation
Move from evaluation tools toward precise definitions.
Precision laboratory
Experiment with ε and δ neighborhoods around a candidate limit.
The lab samples the graph numerically, so it builds intuition for the quantified definition rather than replacing a proof.
LIMIT DEFINITION
Input X
Inspector
f(0.500)1.5000
"f(x) = (x² - 1) / (x - 1). At x=1, the function breaks. But the limit stands."
Proof Engine
Turn on to prove the limit.
Ways a finite two-sided limit can fail
Different failure modes require different diagnoses.
Jump
The left and right limits approach different finite values, so there is no single two-sided limit.
Unbounded
Values grow without bound in magnitude. One-sided signs may agree or disagree, and the ordinary finite limit does not exist.
Oscillation
The function keeps visiting separated output values arbitrarily close to the target input rather than settling toward one value.
Removable hole
The nearby values approach one finite number even though the function is undefined there or assigned a different value.