Limits

A limit describes the value a function approaches as its input moves near a target, whether or not the function is defined there.

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A limit describes the value approached from nearby inputs, whether or not the point is filled there. Left and right curve pieces approach the same open point while arrows show both directions of approach. A separate filled point illustrates that function value and limiting value are distinct questions.

Limits let calculus discuss motion toward a point without requiring the function's value at that point to tell the whole story.

Meaning and notation

The statement

\lim_{x\to a}f(x)=L

means that $f(x)$ can be made as close to $L$ as desired by taking $x$ sufficiently close to $a$ from both sides, without requiring $x=a$.

The limit concerns nearby behaviour. The actual value $f(a)$ may equal $L$, differ from $L$, or be undefined.

Estimating from a table

Choose inputs approaching $a$ from below and above. If the corresponding outputs approach the same value, that value is evidence for the limit.

Use enough precision to see a trend, but remember that a finite table suggests rather than proves a limit. A graph or algebraic argument should support the conclusion.

Estimating from a graph

Trace the graph toward $x=a$ from the left and right. Focus on the approached height, not merely a filled or open point at $x=a$.

An open circle can represent the limiting value even when the function is missing there. A filled point elsewhere at the same input gives $f(a)$ but does not overwrite the nearby approach.

Worked example: remove a common factor

The original function has a hole at $x=3$, yet the limit exists.

Common mistakes

Substituting the point and stopping at $0/0$. It is an indeterminate signal to simplify.

Confusing $f(a)$ with the limit. Nearby behaviour may differ from the point value.

Checking only one side. A two-sided limit requires agreement from both sides.

Treating infinity as an ordinary number. It describes unbounded behaviour.

Cancelling terms instead of factors. Factor algebraically before cancellation.

Quick self-check

What happens under direct substitution?
If indeterminate, can I factor, rationalize, or combine fractions?
Do left- and right-hand behaviours agree?
Am I reporting an approached value rather than blindly reading $f(a)$?
Does a graph or table support the algebra?
Are domain restrictions and asymptotes handled explicitly?