Connecting Multiple Representations of Limits

AP Calculus · Limits and Continuity

Connecting Multiple Representations of Limits

Picture, table, formula, sentence — one limit, and all four must agree.

key idea$\text{One limit, four descriptions — each must give the same answer}$

Four views of one idea

A limit question can arrive as a picture, a table, a formula or a sentence, and the answer must be the same in all four. The exam tests the translation as much as the value, so practise starting in one representation and finishing in another.

Graphical

Numerical

$x$	$1.9$	$1.99$	$2.001$	$2.01$
$f(x)$	$4.9$	$4.99$	$5.001$	$5.01$

Analytical

$f(x)=\dfrac{x^{2}+x-6}{x-2}$

For $x\neq2$ this factors as $\dfrac{(x-2)(x+3)}{x-2}=x+3$, so the limit is $\lim_{x\to2}(x+3)=\mathbf{5}$.

Verbal

“As $x$ gets arbitrarily close to $2$ from either side, $f(x)$ gets arbitrarily close to $5$ — even though $f(2)$ is undefined.”

Translating between views

limit exists, value elsewhere

a jump — three different heights

Left: the two branches meet at height $5$, so the limit is $5$ — while the filled dot puts $f(2)=2$. Right: the branches head for $1$ and $4$, so the limit does not exist, and $g(3)=2$ is a third number again. Left limit, right limit and value are independent.

Each pair of representations has a characteristic move. Learn the move, not just the example.

Starting from	Going to	The move
graph	value	trace each branch; read the height it heads for, ignoring the plotted point
table	value	read down both sides; see what the outputs settle on
formula	value	substitute; if $\tfrac00$, rewrite and substitute again
value	verbal	“as $x$ approaches $c$ from either side, $f(x)$ approaches $L$”
graph	formula	read the holes and asymptotes, then build a function with them
formula	graph	find zeros, holes and asymptotes; sketch the branches around them

Only the first three produce a number. The last three are the ones students practise least — and the ones that appear on connecting-representations questions.

invariants

All four descriptions must agree. A mismatch means an error, not a subtlety.
Graph and table give estimates; the formula gives certainty.
Every representation is silent about $f(c)$ — that is a separate question in all four.
The verbal form must say from either side, or it describes a one-sided limit.

Worked examples

EX 1

Graph in, formula out

graph → analytical

The graph above is a straight line of slope $1$ through $(0,3)$ with a hole at $(2,5)$. Write a formula for such an $f$.

Read the line

Slope $1$ and intercept $3$ give $y=x+3$ everywhere except at $x=2$.

Encode the hole

A hole at $x=2$ needs a factor of $(x-2)$ in both numerator and denominator.

Build it

$f(x)=\dfrac{(x-2)(x+3)}{x-2}$, which equals $x+3$ for $x\neq2$ and is undefined at $2$.

Expand for a tidier form

$f(x)=\dfrac{x^{2}+x-6}{x-2}$.

Check against the picture

Undefined at $2$, and heading for $5$ there. Matches the hole exactly.

EX 2

Table in, verbal out

numerical → verbal

Using the table above, state the limit in words and say what it does not tell you.

Read the left side

$4.9$ then $4.99$ — climbing toward $5$.

Read the right side

$5.01$ then $5.001$ — descending toward $5$.

Write the sentence

“As $x$ approaches $2$ from either side, $f(x)$ approaches $5$.”

State the limitation

The table says nothing about $f(2)$ — no row for it, and a limit would ignore it anyway.

Flag the precision

This is an estimate. Only the factored formula makes $5$ exact.

EX 3

Verbal in, graph out

verbal → graphical

“$g$ approaches $1$ as $x\to3$ from the left, approaches $4$ from the right, and $g(3)=2$.” Describe the graph and give $\lim\limits_{x\to3}g(x)$.

Left branch

It rises or falls toward height $1$ at $x=3$, ending in an open circle at $(3,1)$.

Right branch

It begins at an open circle at $(3,4)$ and continues rightward.

The value

A filled dot sits at $(3,2)$ — on neither branch.

The limit

The one-sided limits are $1$ and $4$; they disagree, so $\lim_{x\to3}g(x)$ does not exist.

Note the three separate facts

Left limit, right limit and value are three independent pieces of information. A jump can carry all three at different heights.

EX 4

Formula in, everything out

analytical → all four

For $h(x)=\dfrac{x^{2}-9}{x-3}$, give the limit at $x=3$ analytically, then describe the graph, the table and the sentence.

Analytical

$\dfrac{(x-3)(x+3)}{x-3}=x+3$ for $x\neq3$, so $\lim_{x\to3}h(x)=6$.

Graphical

The line $y=x+3$ with an open circle at $(3,6)$ and no filled dot.

Numerical

$h(2.9)=5.9$, $h(2.99)=5.99$, $h(3.01)=6.01$ — both sides closing on $6$.

Verbal

“As $x$ approaches $3$ from either side, $h(x)$ approaches $6$, although $h(3)$ is undefined.”

Confirm consistency

All four agree on $6$ and all four agree $h(3)$ does not exist. That agreement is the check.

Watch out

Estimates and exact values are different claims

Graph and table support “approximately”. Only algebra licenses an exact answer — say which you have.

“Approaches” needs a direction

A verbal description without from either side describes a one-sided limit. On a jump that changes the answer completely.

Do not smuggle $f(c)$ into the limit

All four representations are silent about the value at $c$. A filled dot, a missing table row and an undefined formula are each a separate fact.

Disagreement means a mistake

If your graph reading and your algebra differ, one of them is wrong. Consistency across representations is the built-in check.

Course map. Topic 1.9 · The CED assigns this topic no unique learning objective — it is assessed as movement between the graphical, numerical, analytical and verbal representations of a limit.