Lecture 4 · Barnett ch. 3 to 4 · about 45 min

Limits

What a limit means before any derivative is in sight: the value a function approaches, one-sided limits, the algebra that removes 0/0, and what happens as output grows without bound.

By the end you can

Read one-sided limits from a graph and decide when a two-sided limit exists
Apply the limit laws to evaluate limits of sums, products and quotients
Factor to resolve a 0/0 indeterminate form at a removable discontinuity
Evaluate limits at infinity to find horizontal asymptotes, such as long-run average cost

What a limit means

A limit describes what a function approaches as the input gets arbitrarily close to some value, whether or not the function actually reaches that value there. This is a different question from “what is $f(a)$ ?” — a limit can exist even where $f(a)$ does not, and $f(a)$ can exist without the limit existing.

Limit notationFormula sheet →

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

$a$: the input value being approached
$L$: the number f(x) approaches
$x \to a$: x gets arbitrarily close to a, from either side, without ever equaling a

Use it when you need to state or compute what a function approaches near a point, especially one where the function itself may misbehave.

One-sided limits

The left-hand limit looks only at inputs approaching $a$ from below; the right-hand limit, from above. The two-sided limit exists only when both agree.

Existence of a two-sided limitConditionFormula sheet →

\lim_{x \to a} f(x) = L \iff \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L

$x \to a^-$: approach from the left (below a)
$x \to a^+$: approach from the right (above a)

Use it when a piecewise function or a graph has a possible jump, and you need to check whether the limit exists at the seam.

Limit laws

For well-behaved (continuous) functions, limits can be computed by direct substitution, and they combine the way you would hope.

Limit lawsRulesFormula sheet →

\lim_{x \to a}[f(x) \pm g(x)] = \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x) \qquad \lim_{x \to a}[f(x)\,g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)

$f, g$: the two functions being combined, each with an individually existing limit

Use it when a limit is built from sums, differences or products of simpler pieces you already know how to evaluate.

For a quotient, the same rule holds provided the limit of the denominator is not zero: $\lim \frac{f}{g} = \frac{\lim f}{\lim g}$ when $\lim g \neq 0$ . When the denominator’s limit is zero, direct substitution fails and more work is needed — the next section handles exactly this case.

Removable discontinuities: the 0/0 form

If direct substitution gives $\frac{0}{0}$ , the function is not necessarily undefined in the limit — it usually means a common factor can be cancelled first.

Try it: approach from both sides

f(x) = \dfrac{x^2 - 4}{x - 2}

Blue point approaches from the left, orange from the right, as h shrinks.

Step size h1.000

h	f(a − h)	f(a + h)
1	3.000	5.000
0.5	3.500	4.500
0.1	3.900	4.100
0.01	3.990	4.010
0.001	3.999	4.001

Both sides settle on the same number: $\lim_{x \to a} f(x) = 3.999$ .

Switch between the three cases. The hole shows a limit that exists even though the function is undefined there. The jump shows a limit that fails to exist because the two sides disagree, even though the function itself is perfectly defined at that point. The asymptote shows a limit that fails to exist because the function grows without bound.

Limits at infinity

Letting $x$ grow without bound answers questions like: what happens to average cost as output becomes very large? Divide numerator and denominator by the highest power of $x$ present.

Limits at infinityFormula sheet →

\lim_{x \to \infty} \frac{a_n x^n + \dots}{b_m x^m + \dots} = \begin{cases} 0 & n < m \\ a_n/b_m & n = m \\ \pm\infty & n > m \end{cases}

$n, m$: the degrees of the numerator and denominator
$a_n, b_m$: their leading coefficients

Use it when a rational function's behaviour as x becomes very large is needed, such as long-run average cost.

Exam practice

Summary and review

$\lim_{x \to a} f(x) = L$ describes what $f$ approaches near $a$ , independent of $f(a)$ itself.
A two-sided limit exists only when the left-hand and right-hand limits agree.
Limit laws let you compute limits of sums, differences, products termwise.
A $0/0$ result means simplify (usually factor and cancel) before substituting again.
For limits at infinity of a rational function, compare the degrees of numerator and denominator.
Long-run average cost approaching a constant is a limit at infinity in disguise.