Defining Limits and Using Limit Notation

AP Calculus · Limits and Continuity

Defining Limits and Using Limit Notation

What the function approaches — read from both sides, and independent of the value at the point.

key idea$\lim_{x\to c}f(x)=L:\ \text{what }f\text{ APPROACHES, not what it equals}$

What it says

The limit of a function

\[\lim_{x\to c}f(x)=L\]

As $x$ gets arbitrarily close to $c$ — from both sides, and without ever equalling $c$ — the values $f(x)$ get arbitrarily close to the single number $L$. Everything in that sentence matters: both sides, arbitrarily close, and a single finite number.

What the definition deliberately ignores

The value $f(c)$ plays no part whatsoever. The function may be undefined at $c$, or defined with a completely different value, and the limit is unaffected. The limit describes the approach, never the arrival.

Statement	True or false	Why
$\lim_{x\to c}f(x)$ exists $\Rightarrow f(c)$ exists	false	$f$ can have a hole at $c$ and still have a limit there
$f(c)$ exists $\Rightarrow\lim_{x\to c}f(x)$ exists	false	a jump discontinuity has a value but no two-sided limit
Both exist and are equal	this is continuity	precisely the definition of continuity
$\lim_{x\to c}f(x)=\infty$ means the limit exists	false	$\infty$ is not a number; the limit does not exist, though the notation describes how it fails

The first two rows are independent facts. Neither existence implies the other — which is exactly why continuity needs its own definition.

Reading the notation

$\lim_{x\to c}f(x)=L$The two-sided limit. Requires the left and right limits to both exist and agree. This is the default meaning when no superscript appears.

$\lim_{x\to c^{-}}f(x)$The left-hand limit: $x$ approaches $c$ through values smaller than $c$. The minus refers to the side, never to a negative number.

$\lim_{x\to c^{+}}f(x)$The right-hand limit: $x$ approaches through values larger than $c$. Again a side, not a sign.

$\lim_{x\to\infty}f(x)=L$End behaviour: as $x$ grows without bound, $f(x)$ approaches $L$. This corresponds to a horizontal asymptote $y=L$.

$\lim_{x\to c}f(x)=\infty$A description of failure: $f$ grows without bound near $c$. The limit does not exist, and this notation records why. Corresponds to a vertical asymptote.

“DNE”Does not exist. Used when the two sides disagree, or the function oscillates without settling. Prefer $\pm\infty$ when the failure is unbounded growth — it says more.

Written in symbols	Read aloud as
$\lim_{x\to 3}f(x)=7$	“the limit of $f$ of $x$, as $x$ approaches $3$, equals $7$”
$\lim_{x\to 3^{-}}f(x)=2$	“… as $x$ approaches $3$ from the left …”
$\lim_{x\to 0^{+}}\ln x=-\infty$	“… from the right, decreases without bound”
$\lim_{x\to\infty}\tfrac1x=0$	“… as $x$ increases without bound, approaches $0$”

Saying it aloud correctly is examinable in its own right — the CED lists communication and notation as a mathematical practice.

Worked examples

EX 1

The limit ignores the function's value

hole

Let $f(x)=\dfrac{x^{2}-1}{x-1}$. Find $\lim_{x\to1}f(x)$ and compare it with $f(1)$.

Try the value

$f(1)=\dfrac{0}{0}$ — undefined. So $f(1)$ does not exist.

Study the approach instead

For every $x\neq1$, $\dfrac{(x-1)(x+1)}{x-1}=x+1$. The cancellation is legitimate precisely because $x$ never equals $1$ in a limit.

Take the limit

As $x\to1$, $x+1\to 2$. So $\lim_{x\to1}f(x)=2$.

State the contrast

The limit is $2$; the function value does not exist. This is the clearest demonstration that the two are independent.

EX 2

Both sides must agree

one-sided

For $g(x)=\begin{cases}x+1,& x\lt 2\\ 5-x,& x\ge 2\end{cases}$, evaluate the one-sided limits at $x=2$ and then the two-sided limit.

Left-hand limit

Approaching from below uses the first piece: $\lim_{x\to2^{-}}g(x)=2+1=3$.

Right-hand limit

Approaching from above uses the second: $\lim_{x\to2^{+}}g(x)=5-2=3$.

Compare

Both one-sided limits equal $3$, so they agree and $\lim_{x\to2}g(x)=3$.

Check the value separately

$g(2)=5-2=3$ as well, so $g$ is in fact continuous here — but that was a separate check, made after the limit.

EX 3

Express one limit three ways

multiple representations

A function $h$ satisfies $\lim_{x\to4}h(x)=6$. Express this graphically, numerically and verbally.

Graphically

Near $x=4$ the graph of $h$ approaches the height $6$ from both directions. There may be a hole at $(4,6)$, or a point plotted elsewhere — the approach is what the picture shows.

Numerically

Values of $h$ at $x=3.9,\,3.99,\,3.999$ and at $x=4.1,\,4.01,\,4.001$ both close in on $6$. A table converging from both sides is the numerical statement.

Verbally

“As $x$ gets arbitrarily close to $4$ from either side, the values of $h$ get arbitrarily close to $6$.”

Why this matters

The CED asks you to move between these representations, not merely to compute. A limit question can be posed in any of the three and answered in any other.

EX 4

A limit that fails to exist

non-example

For $p(x)=\dfrac{\lvert x\rvert}{x}$, explain why $\lim_{x\to0}p(x)$ does not exist.

Right side

For $x\gt 0$, $\lvert x\rvert=x$, so $p(x)=1$ and $\lim_{x\to0^{+}}p(x)=1$.

Left side

For $x\lt 0$, $\lvert x\rvert=-x$, so $p(x)=-1$ and $\lim_{x\to0^{-}}p(x)=-1$.

Apply the definition

The definition demands the values approach a single number from both sides. Here they approach $1$ and $-1$.

State it correctly

$\lim_{x\to0}p(x)$ does not exist. Note that both one-sided limits do exist — failure of the two-sided limit does not mean nothing exists.

Boundary cases

The superscript is a side, not a sign

In $\lim_{x\to3^{-}}$ the minus means “from the left”. It does not mean $x\to-3$, and it says nothing about whether $f$ is negative.

$=\infty$ is a description of failure

Writing $\lim_{x\to c}f(x)=\infty$ is standard and informative, but the limit does not exist — $\infty$ is not a number. Say “increases without bound” when asked to explain.

Never substitute $x=c$ to justify a limit

Substitution often gives the right number, but only because of continuity, which is a later theorem. On a justification question, appeal to the approach or to a limit law, not to plugging in.

One-sided limits can exist when the two-sided one does not

As in Example 4. “The limit does not exist” is a claim about the two-sided limit only — be precise about which limit you mean.

Course map. Topic 1.2 · LOs LIM-1.A (represent limits analytically using correct notation) and LIM-1.B (interpret limits expressed in analytic notation) — EK LIM-1.A.1, LIM-1.B.1: a limit can be expressed in multiple ways, including graphically, numerically and analytically.