Lecture 4 · Barnett 2-1 to 2-2 · about 45 min

Limits

What a limit is, reading one-sided limits from a graph, the algebra for 0/0 (factor, rationalise), infinite limits and vertical asymptotes, and limits at infinity with horizontal asymptotes such as long-run average cost.

By the end you can

Read one-sided and two-sided limits from a graph, separately from the function value
Evaluate limits by substitution, factoring and rationalising, including piecewise and absolute value cases
Find infinite limits and vertical asymptotes from the sign of the denominator
Evaluate limits at infinity and find horizontal asymptotes

Key idea · Exam recipe (memorise this)

Substitute $x = a$ first. A number → that is the limit.
$\frac{c}{0}$ with $c \ne 0$ → infinite limit. Find the sign of the denominator on each side: $+\infty$ or $-\infty$ . Different signs → two-sided limit does not exist; vertical asymptote $x = a$ .
$\frac{0}{0}$ → simplify, then substitute again: factor and cancel; with a square root multiply by the conjugate; with fractions inside combine them.
Piecewise at the boundary: left limit from the left formula, right limit from the right formula. Equal → that is the limit; different → does not exist.
$\frac{|x - a|}{x - a}$ : $-1$ from the left, $+1$ from the right → does not exist.
$x \to \pm\infty$ (rational function): divide by the highest power of $x$ in the denominator. Top degree smaller → 0; equal → ratio of leading coefficients; bigger → $\pm\infty$ .
From a graph: follow the curve to $a$ from each side; the dot at $a$ is $f(a)$ , not the limit.

How the exam asks it: three core exam templates

In midterm exams, limit questions focus on three canonical structures: evaluating one-sided limits and piecewise boundaries from graphs or formulas, resolving indeterminate $0/0$ forms via factoring, conjugates, or fraction simplification, and analyzing infinite limits, vertical asymptotes, and long-run asymptotic behaviour. Master these three worked solutions, then follow the cheat-sheet and 4-step method below.

Exam template 1: Piecewise boundaries, one-sided limits, and function values

Question: Consider the piecewise function: $f(x) = \begin{cases} 2x + 1 & \text{if } x \lt 2 \\ x^2 & \text{if } x \ge 2 \end{cases}$ a) Find the left-hand limit $\lim_{x \to 2^-} f(x)$ .
b) Find the right-hand limit $\lim_{x \to 2^+} f(x)$ .
c) State whether the two-sided limit $\lim_{x \to 2} f(x)$ exists, and give the mathematical reason.
d) Evaluate the function value $f(2)$ .

Exam template 2: Indeterminate 0/0 limits (factoring, conjugate, and complex fractions)

Question: Evaluate the following limits, showing all algebraic steps:
a) $\lim_{x \to 2} \dfrac{x^2 - 4}{x - 2}$
b) $\lim_{x \to 4} \dfrac{\sqrt{x} - 2}{x - 4}$
c) $\lim_{h \to 0} \dfrac{\frac{1}{2 + h} - \frac{1}{2}}{h}$

Exam template 3: Infinite limits, vertical asymptotes, and long-run average cost

Question: Consider the functions $f(x) = \dfrac{x + 2}{x - 1}$ and business average cost $\bar{C}(x) = \dfrac{5000 + 20x}{x}$ .
a) Evaluate one-sided limits $\lim_{x \to 1^-} f(x)$ and $\lim_{x \to 1^+} f(x)$ , and identify the equation of the vertical asymptote.
b) Evaluate the long-run limit $\lim_{x \to \infty} \bar{C}(x)$ and state the equation of the horizontal asymptote.
c) Provide the economic interpretation of the horizontal asymptote $y = 20$ .

Cheat-sheet & the 4-step method

· Cheat-sheet: limit evaluation techniques and asymptotic behaviour

Limit Form	Identification	Required Method	How to write the exam answer
Direct substitution	Denominator $\ne 0$	Plug in $x = a$ directly	State substituted value, e.g. $\lim = f(a)$
Indeterminate $\dfrac{0}{0}$ (polynomial)	Both top and bottom equal $0$	Factor $(x - a)$ out of numerator and denominator, cancel	Cancel factor, re-substitute $x = a$ , state finite number
Indeterminate $\dfrac{0}{0}$ (radical)	Contains $\sqrt{g(x)} - c$	Multiply numerator & denominator by conjugate $(\sqrt{g(x)} + c)$	Expand $(A-B)(A+B) = A^2-B^2$ , cancel $(x-a)$ , substitute
Indeterminate $\dfrac{0}{0}$ (nested fraction)	Contains $\dfrac{1}{a+h} - \dfrac{1}{a}$	Find common denominator on numerator, combine	Divide by $h$ , substitute $h = 0$
Infinite limit $\dfrac{c}{0}$ ( $c \ne 0$ )	Top $\ne 0$ , bottom $= 0$	Sign analysis from left ( $0^-$ ) and right ( $0^+$ )	State $-\infty$ or $+\infty$ ; conclude vertical asymptote $x = a$
Piecewise boundary	Function rule changes at $x = a$	Test $\lim_{x \to a^-}$ and $\lim_{x \to a^+}$ separately	Equal $\implies$ limit exists; unequal $\implies$ DNE
Limit at infinity $\dfrac{P(x)}{Q(x)}$	$x \to \infty$ or $x \to -\infty$	Compare polynomial degrees of top and bottom	Equal degrees: ratio of leading coefficients; horizontal asymptote $y = a_n/b_m$

Key idea · The 4-step method for any exam question on limits

Step 1: Test by direct substitution ( $x = a$ ).
- Finite real number $\implies$ task complete, that number is the limit.
- Non-zero over zero ( $c/0$ , $c \ne 0$ ) $\implies$ infinite limit; prepare for sign testing ( $+\infty$ or $-\infty$ ).
- Zero over zero ( $0/0$ ) $\implies$ indeterminate form; algebraic reduction required.
Step 2: Choose and apply the appropriate algebraic tool.
- Polynomials $\implies$ factor difference of squares or trinomials, cancel $(x - a)$ .
- Square roots $\implies$ multiply top and bottom by conjugate $(\sqrt{u} + c)$ .
- Internal fractions $\implies$ find common denominator and cancel denominator $h$ .
Step 3: Evaluate the reduced expression.
- Substitute $x = a$ (or $h = 0$ ) into the simplified expression.
- For piecewise functions, evaluate left and right rules independently and check equality.
Step 4: State the bold exam answer and run a quick verification.
- Clearly state whether the limit is a finite number, $+\infty$ , $-\infty$ , or “Does Not Exist” (DNE).
- Test with a calculator value very close to $a$ (e.g. $a \pm 0.01$ ) to confirm sign and magnitude.

What a limit means

$\lim_{x \to a} f(x) = L$ means: as $x$ gets close to $a$ (from both sides, but never equal to $a$ ), $f(x)$ gets close to $L$ . The value $f(a)$ itself plays no role: it may be different, or not exist.

Limit notationFormula sheet →

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

$a$: the input value approached
$L$: the number f(x) approaches
$x \to a$: x close to a, from either side, never equal to a

Use it when you need what a function approaches near a point, especially where it misbehaves.

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

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

Use it when a graph or a piecewise function may jump at a, and you must decide whether the limit exists.

Limits from a graph

The most common exam format: a graph with open and closed dots and five or six questions about it.

Substitution and the limit laws

Limits split over sums, differences, products and quotients (if the denominator’s limit is not 0). For polynomials, and for rational functions where the denominator is not 0, just substitute.

Limit lawsRulesFormula sheet →

\lim[f \pm g] = \lim f \pm \lim g, \qquad \lim[f g] = \lim f \cdot \lim g, \qquad \lim \frac{f}{g} = \frac{\lim f}{\lim g}\ (\lim g \ne 0)

$f, g$: functions whose limits at a exist

Use it when a limit is built from pieces whose limits you know.

Piecewise functions and absolute value

The 0/0 form: factor or rationalise

$\frac{0}{0}$ after substitution is indeterminate: it says nothing yet. Remove the common factor, then substitute again.

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: $\lim_{x \to a} f(x) = 3.999$ .

Infinite limits and vertical asymptotes

If substitution gives $\frac{c}{0}$ with $c \ne 0$ , the function blows up. The sign of the tiny denominator decides $+\infty$ or $-\infty$ .

$\lim_{x\to2^+} \frac{1}{x-2} = +\infty$ (tiny positive denominator), $\lim_{x\to2^-} \frac{1}{x-2} = -\infty$ (tiny negative). Vertical asymptote $x = 2$ ; two-sided limit does not exist. As $x \to \pm\infty$ , $y \to 0$ : horizontal asymptote $y = 0$ .

Limits at infinity

Rational function as x → ±∞Formula sheet →

\lim_{x \to \pm\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}

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

Use it when you need the long-run behaviour or the horizontal asymptote of a rational function.

Exam practice

Summary and review

The limit is about the approach, not the value at $a$ ; it exists only if both one-sided limits agree.
Substitute first. $\frac00$ : factor, rationalise, or combine fractions. $\frac{c}{0}$ : infinite limit, check signs.
Piecewise: each side uses its own formula. $\frac{|x-a|}{x-a}$ : $\pm 1$ , no limit.
$x \to \pm\infty$ : compare degrees; equal degrees give the horizontal asymptote $y = a_n/b_m$ .