CALCULUS · FOUNDATION

Limits & continuity

A limit describes the value a function approaches as its input approaches a point. It lets us reason precisely about local behavior — even when direct substitution fails.

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

Inputs near \(a\) force outputs near \(L\).

1. What a limit means

The statement \(\lim_{x\to a}f(x)=L\) concerns values of \(f(x)\) for \(x\) near \(a\). The value \(f(a)\) may equal \(L\), differ from \(L\), or not exist at all.

Key distinction. A limit describes nearby behavior. A function value describes what happens exactly at one input.
limit Lf(a)

2. The precise ε–δ definition

We write \(\lim_{x\to a}f(x)=L\) if for every \(\varepsilon>0\) there exists a \(\delta>0\) such that

\[0<|x-a|<\delta\quad\Longrightarrow\quad |f(x)-L|<\varepsilon.\]

In words: no matter how small an output tolerance \(\varepsilon\) is requested, we can choose an input tolerance \(\delta\) that guarantees the function stays within it.

Why exclude \(x=a\)?

The condition uses \(0<|x-a|\), so the limit depends only on nearby points. That is exactly why removable holes can still have limits.

3. Limit laws

If \(\lim f(x)=A\) and \(\lim g(x)=B\), then — whenever the expressions are defined —

\[\lim(f+g)=A+B,\quad \lim(cf)=cA,\quad \lim(fg)=AB,\quad \lim\frac{f}{g}=\frac AB\;(B\neq0).\]

Polynomials are continuous everywhere, so their limits can be found by direct substitution. Rational functions are continuous wherever the denominator is nonzero.

Two-sided limitExists only when left and right limits both exist and are equal.
Infinite limit\(f(x)\to\pm\infty\) describes unbounded growth, not a finite real limit.
Limit at infinity\(x\to\pm\infty\) describes long-run behavior, often revealing horizontal asymptotes.

4. Continuity

A function is continuous at \(a\) when three conditions hold:

\[f(a)\text{ exists},\qquad \lim_{x\to a}f(x)\text{ exists},\qquad \lim_{x\to a}f(x)=f(a).\]

Common discontinuities include removable holes, jump discontinuities and infinite discontinuities.

Intermediate Value Theorem. If \(f\) is continuous on \([a,b]\), then it takes every value between \(f(a)\) and \(f(b)\). This is why sign changes can guarantee roots.

5. Worked examples

Example A — direct substitution

\[\lim_{x\to2}(x^2+3x)=4+6=10.\]

Example B — removable discontinuity

Direct substitution in \((x^2-1)/(x-1)\) gives \(0/0\), an indeterminate form. Factor first:

\[\frac{x^2-1}{x-1}=\frac{(x-1)(x+1)}{x-1}=x+1\quad(x\neq1),\]
\[\therefore\quad \lim_{x\to1}\frac{x^2-1}{x-1}=2.\]

Example C — one-sided mismatch

If \(f(x)=0\) for \(x<0\) and \(f(x)=1\) for \(x\ge0\), then

\[\lim_{x\to0^-}f(x)=0,\qquad \lim_{x\to0^+}f(x)=1.\]

Because the one-sided limits differ, the two-sided limit does not exist.

6. Common mistakes

Calling \(0/0\) the answer. It is an indeterminate form signaling that more algebra or analysis is needed.
Confusing the limit with the function value. They agree only when the function is continuous at that point.
Ignoring one-sided behavior. A two-sided limit requires matching left and right limits.
Treating infinity as a real number. Infinite limits describe unbounded behavior rather than convergence to a finite value.

7. Practice

  1. Find \(\lim_{x\to3}(2x^2-x)\).
  2. Find \(\lim_{x\to2}(x^2-4)/(x-2)\).
  3. Does \(f(x)=|x|/x\) have a limit as \(x\to0\)?
Show answers

1) \(15\). 2) Factor to \(x+2\), so the limit is \(4\). 3) No: the left-hand limit is \(-1\), while the right-hand limit is \(1\).

8. Continue learning

Derivatives →Integrals →Review functions →