Limits, continuity and existence

What this concept is

A limit describes nearby values, not necessarily the value at the point. A two-sided limit exists only when both one-sided limits agree. Continuity at a requires that limit to equal f(a).

Why it matters here

Derivatives and integrals rely on limiting arguments. Sampling a few inputs can suggest a limit but cannot prove it.

Concrete example

For x≠3, (x²−9)/(x−3)=x+3, so its limit at 3 is 6 even though the original expression is undefined there. Defining f(3)=6 repairs the hole. In contrast, |x|/x tends to −1 from the left and 1 from the right at zero: no two-sided limit.

Common confusion / misconception

An expression 0/0 is an indeterminate form, not the number zero. Cancellation at nearby inputs does not make the original denominator valid at its zero.

How to use it

Try continuity and substitution first; then simplify on a punctured neighborhood or compare one-sided behavior. Precisely, a finite limit L at a means: for every ε>0 there is δ>0 such that 0<|x−a|<δ implies |f(x)−L|<ε. For f(x)=2x+1 at a=3 choose δ=ε/2. This establishes a guarantee for every nearby input, unlike a table.

Check yourself

Does the function equal to x for x≠0 and 7 at x=0 have a limit at zero? Is it continuous? Explain both.

Mini drill or application

Prove the limit of 5x−2 at x=1 is 3 using δ=ε/5. Give a function with unequal one-sided limits. State the intermediate value theorem: a continuous function on [a,b] takes every value between f(a) and f(b); identify why continuity matters for root finding.

Read this only if stuck

OpenStax selective support. Use the reading guide for the section sequence and practice contract. Record your reasoning before consulting solutions.