Defining Limits & Limit Notation | AP Calculus Topic 1.2

1. What is a limit?

A limit describes the value a function approaches as its input approaches a specified number. The key question is what happens near that input, rather than what happens only at the input itself.

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

This says that the outputs $f (x)$ can be made as close as we want to the real number $L$ by taking inputs $x$ sufficiently close to $a$ , while keeping $x \neq a$ . For an ordinary two-sided limit at an interior input, approach from both sides.

Nearby behavior is the focus.

The function may have the same value as the limit, a different value, or no defined value at the target. Changing a single function value does not change the nearby limiting behavior.

Topic 1.2 in AP® Calculus AB and BC focuses on writing limits correctly and interpreting them across formulas, tables, graphs and words. A formal epsilon–delta proof is not required for the AP course. The extended notation and examples below also preview later Unit 1 topics.

If you need the motivation for limits, review Topic 1.1: can change occur at an instant?, where nearby average rates lead to an instantaneous rate.

2. Read and write limit notation

Read $lim_{x \to a} f (x) = L$ as “the limit of f of x, as x approaches a, is L.” The input approaches the target; the output approaches the limiting value.

The four parts of a limit statement
Part of the statement	How to read it	What it tells you
$lim$	Limit	We are describing approaching behavior.
$x \to a$	As the input approaches the target	The target is an input value, not an output.
$f (x)$	The function output	These values are examined near the target.
$= L$	Equals the limiting value	This equality describes the limit, not necessarily the function value at the target.

An arrow is not an equals sign

$x \to a$ describes a process of approach. $x = a$ names the exact input. Do not replace the arrow under the limit with an equals sign.

The letter used for the input is not important. For example,

lim_{t \to 3} g (t) = 11

means that the outputs of the function named g approach eleven when its input approaches three. Changing a dummy input letter is fine; changing the target number, function or approach direction changes the statement.

3. Left-hand and right-hand limits

Approach from the left

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

Use inputs less than the target: $x < a$ . Read the superscript minus as “from the left.”

Approach from the right

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

Use inputs greater than the target: $x > a$ . Read the superscript plus as “from the right.”

The superscript describes the side of the input target. It does not say that the inputs or outputs must be negative or positive. Approaching five from the left can use positive inputs such as 4.9 and 4.99. Approaching negative two from the right can use negative inputs such as negative 1.9.

Translate a one-sided statement into words

lim_{x \to 5^{-}} f (x) = 3

Read: “As x approaches five from values less than five, f of x approaches three.” It does not tell you the function value at five or what happens to the right of five. Inputs like 4.9 and 4.99 illustrate the approach; a limit concerns all sufficiently close permitted inputs, not just those two samples.

4. When does a two-sided limit exist?

When the function has inputs arbitrarily close to the target on both sides, a finite two-sided limit exists precisely when both one-sided limits approach the same finite number:

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

In that case, $lim_{x \to a} f (x) = L$ . The function value at the target is a separate question.

Determine the left-hand limiting behavior.
Determine the right-hand limiting behavior.
If both approach one finite number, use that number as the two-sided limit.
If the finite one-sided limits differ, the two-sided limit does not exist. If they are distinct finite numbers, this is a jump.
If either side is unbounded or fails to settle, no finite two-sided limit exists. Describe the behavior more specifically when possible.

Two finite unequal limits give a jump; not every failed limit is a jump. At a domain endpoint, only one side may be available. State that you are using a one-sided limit rather than inventing missing values outside the domain.

5. Connect formulas, tables and graphs

Limits can be represented analytically with an expression, numerically with nearby values, and graphically with a curve. A verbal interpretation explains what the notation means.

Analytical representation

For the continuous linear function $f (x) = 2 x + 5$ , direct substitution gives

lim_{x \to 3} (2 x + 5) = 2 (3) + 5 = 11

Numerical representation

Linear example: outputs approach eleven from both sides
Input from the left	Output	Input from the right	Output
$2.9$	$10.8$	$3.1$	$11.2$
$2.99$	$10.98$	$3.01$	$11.02$
$2.999$	$10.998$	$3.001$	$11.002$

The table supports $lim_{x \to 3} f (x) = 11$ . The formula establishes why the outputs keep approaching eleven. A finite table alone cannot prove a limit for an arbitrary unknown function; unlisted values may behave differently.

Graphical representation

Trace the curve toward the target input from each side and read the output heights approached. A filled dot answers the function-value question. An open circle marks an excluded point on that branch; the function may still be defined at the same input at another height.

Use matching distances on each side in a table to compare trends conveniently, then try progressively smaller distances. Symmetric sampling helps organization; it does not guarantee accuracy or detect every oscillation. Continue with Topic 1.3: estimating limits from graphs for more visual practice.

6. A limit is different from a function value

Function value versus limit
Question	Notation	What to inspect
What is the actual output at the target?	$f (a)$	The definition at that input, or the filled point on a graph.
What output is approached nearby?	$lim_{x \to a} f (x)$	The behavior as permitted inputs approach the target.

A limit can exist when the function is undefined at the target or has a different value there. A hole is compatible with a limit; a jump or oscillation is not compatible with a finite two-sided limit. “A break in the graph” alone is too vague to decide.

Worked example 1: a limit exists at a hole

f (x) = \frac{x^{2} - 4}{x - 2}, x \neq 2

Factor the numerator, then cancel the common factor only for inputs where it is nonzero:

\frac{x^{2} - 4}{x - 2} = \frac{(x - 2) (x + 2)}{x - 2} = x + 2

lim_{x \to 2} \frac{x^{2} - 4}{x - 2} = lim_{x \to 2} (x + 2) = 4

The original function has no value at two because its denominator is zero. The expression $\frac{0}{0}$ is not a number. Cancelling does not restore the excluded input, but it reveals the nearby behavior.

Two graphs of y equals x plus 2 near input 2. Both have an open point at (2,4); the second has a filled point at (2,7), showing the same limit with a different function value. — Both graphs approach four at input two. The left function is undefined there; the right function has been separately assigned the value seven. Neither point choice changes the limit.

If we separately assign $f (2) = 7$ , the limit still equals four. Assigning the value four instead would fill the hole and make this function continuous at two. This is a preview of removable discontinuities.

7. A piecewise function with unequal limits

Worked example 2: a jump at input one

f (x) = {\begin{cases} x^{2} & if x < 1 \\ 2 x & if x \geq 1 \end{cases}

For the left-hand limit, use the branch whose condition includes nearby inputs less than one. For the right-hand limit, use the branch for nearby inputs greater than one.

lim_{x \to 1^{-}} f (x) = 1^{2} = 1

lim_{x \to 1^{+}} f (x) = 2 (1) = 2

Since $1 \neq 2$ , the two-sided limit does not exist. Nevertheless, $f (1) = 2$ because the second branch includes equality.

Piecewise graph with x squared to the left of 1 ending in an open point (1,1), and 2x on and to the right of 1 beginning at a filled point (1,2). — The branch approached from the left tends to one; the branch approached from the right tends to two. The filled endpoint gives the function value, not a common limit.

The equality condition selects the point value.

For a one-sided limit, use the rule valid on that side near the target. Do not automatically use the branch containing the equals sign for both limits.

8. DNE, infinite behavior and oscillation

DNE means “does not exist.” When discussing a finite real limit, common reasons are unequal finite one-sided limits, unbounded behavior and persistent oscillation.

Infinite limits describe unbounded outputs

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

lim_{x \to a} f (x) = - \infty

These statements describe outputs increasing without bound or decreasing without bound. Infinity is not a real number or a function value. We say no finite limit exists, while the infinite-limit notation records the specific behavior.

A vertical asymptote at $x = a$ can be established by an appropriate one-sided infinite limit; both sides need not tend in the same direction.

Correctly read a reciprocal near its vertical asymptote

lim_{x \to 2^{-}} \frac{1}{x - 2} = - \infty

lim_{x \to 2^{+}} \frac{1}{x - 2} = \infty

From the left, the denominator is a small negative number; from the right, it is a small positive number. Therefore the two-sided limit does not exist. It is incorrect to assign positive infinity to its two-sided limit.

For comparison, squaring the denominator makes both sides positive:

lim_{x \to 2} \frac{1}{{(x - 2)}^{2}} = \infty

Oscillation can also prevent a limit

The function $\sin (\frac{1}{x})$ has no limit as the input approaches zero. With angles in radians, its outputs keep oscillating rather than settling near one number. A sparse table can miss this behavior. Oscillation itself is not enough to rule out every limit: an oscillation whose amplitude shrinks to zero may still have a limit.

9. Limits at infinity and end behavior

This is an extended notation preview of later Unit 1 material. Limits at infinity describe inputs moving without bound, unlike infinite limits, which describe outputs moving without bound.

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

lim_{x \to - \infty} f (x) = L

A finite end-behavior limit gives a horizontal asymptote $y = L$ in that direction. The two ends can have different limits. A curve may cross a horizontal asymptote; it describes end behavior rather than a barrier.

Worked example 3: a rational function at infinity

lim_{x \to \infty} \frac{3 x^{2} + 2}{5 x^{2} + 4 x + 1}

Divide every term in the numerator and denominator by the highest power, $x^{2}$ , for nonzero inputs:

\frac{3 x^{2} + 2}{5 x^{2} + 4 x + 1} = \frac{3 + \frac{2}{x^{2}}}{5 + \frac{4}{x} + \frac{1}{x^{2}}}

The fractions with growing denominators approach zero. The denominator approaches five, so the quotient approaches

lim_{x \to \infty} \frac{3 + \frac{2}{x^{2}}}{5 + \frac{4}{x} + \frac{1}{x^{2}}} = \frac{3 + 0}{5 + 0 + 0} = \frac{3}{5}

Useful degree comparisons for rational functions

If numerator and denominator have the same degree, the end-behavior limit is the ratio of leading coefficients.
If the numerator has lower degree, the end-behavior limit is zero.
If the numerator has higher degree, there is no finite end-behavior limit; the sign of unbounded behavior depends on the coefficients, degree difference and direction of approach. Do not choose a sign without checking.

For instance, $lim_{x \to \infty} x = \infty$ , while $lim_{x \to - \infty} x = - \infty$ . A single unsigned infinity symbol cannot communicate both directions.

10. Complete notation and special forms

Limit notation at a glance
Notation	Read as	Interpretation
$lim_{x \to a} f (x) = L$	Two-sided finite limit	Nearby outputs from both sides approach the same finite value.
$lim_{x \to a^{-}} f (x) = L$	Left-hand limit	Inputs approach from below the target.
$lim_{x \to a^{+}} f (x) = L$	Right-hand limit	Inputs approach from above the target.
$lim_{x \to \infty} f (x) = L$	Limit at positive infinity	Finite limiting output as the input grows without bound.
$lim_{x \to - \infty} f (x) = L$	Limit at negative infinity	Finite limiting output as the input decreases without bound.
$lim_{x \to a} f (x) = \infty$	Positive infinite limit	Outputs exceed every positive bound near the target.
$lim_{x \to a} f (x) = - \infty$	Negative infinite limit	Outputs decrease below every negative bound near the target.
$lim_{x \to a} f (x) DNE$	No finite limit	State whether the reason is unequal sides, unbounded behavior or oscillation.

Special forms to recognize

Continuity: $lim_{x \to a} f (x) = f (a)$ means the limit agrees with the defined function value. Together with existence of that value and a finite limit, this is continuity at an interior point.
Indeterminate forms: $\frac{0}{0}$ and $\frac{\infty}{\infty}$ do not determine a quotient limit. They describe a limiting form, not legal arithmetic with zero or infinity. Use an appropriate method such as factoring, rewriting or dividing by a power.
Removable discontinuity: a finite limit exists but the function value is missing or differs from it.
Jump: finite left and right limits exist and are unequal.
Vertical asymptote: at least one side has infinite limiting behavior.
Horizontal asymptote: a finite limit at one end gives the corresponding horizontal line.

11. Tips, shortcuts and common mistakes

Separate the input target from the output limit. A target of three does not mean the output limit is three.
Keep the arrow under the limit. Use $x \to a$ , not $x = a$ .
Check both sides for a piecewise function. A defined function value cannot force the one-sided limits to agree.
Read dots carefully. Trace the curve for a limit and the filled dot for a function value. An open point on one branch need not mean the whole function is undefined at that input.
Use substitution where continuity justifies it. Polynomials are continuous everywhere. Rational functions are continuous where their denominators are nonzero; trigonometric functions must also be considered within their domains.
Do not treat an indeterminate form as an answer. Factoring may remove a common factor for nearby inputs, but it does not change the original domain.
Check signs near a zero denominator. “Nonzero over zero” is not a computed value or sufficient evidence for a two-sided positive-infinity limit. For rational functions, study the sign on each side.
Describe infinite behavior precisely. Use the correct one-sided or two-sided statement when justified; do not write positive and negative infinity together as though they were one common limit.
Do not overtrust a table. Smaller distances can strengthen an estimate, but finite samples cannot exclude hidden jumps, oscillation or numerical error.

A concise written justification

State the left-hand value, the right-hand value and whether they agree. If they agree at a finite number, state the two-sided limit. If they do not, say why no finite two-sided limit exists. Keep the function value separate.

12. Quick reference and reading practice

Quick writing guide
You want to…	Write this	Remember
State a finite two-sided limit	$lim_{x \to a} f (x) = L$	Both sides must approach the same finite value.
State a left-hand limit	$lim_{x \to a^{-}} f (x) = L$	Approach using smaller inputs.
State a right-hand limit	$lim_{x \to a^{+}} f (x) = L$	Approach using larger inputs.
Say no finite limit exists	$lim_{x \to a} f (x) DNE$	Explain the behavior that prevents it.
Describe a vertical asymptote	$lim_{x \to a^{+}} f (x) = \infty$	This is one example; inspect the actual side and sign.
Describe a horizontal asymptote	$lim_{x \to \infty} f (x) = L$	The input, rather than the output, approaches infinity.

Read a limit at infinity

lim_{x \to \infty} \frac{1}{x} = 0

As the input becomes arbitrarily large and positive, the reciprocal approaches zero. It stays positive for these inputs; the limit can still be zero even though no finite positive input makes the reciprocal exactly zero.

Read a one-sided infinite limit

lim_{x \to 2^{+}} \frac{1}{x - 2} = \infty

Approach two using inputs greater than two. The outputs increase without bound. This statement says nothing about a finite function value at two and does not describe the left-hand behavior.

Before moving on, make sure you can translate a limit statement into words, distinguish a left-hand limit from a right-hand limit, and explain why a limit and a function value can differ.

13. Continue through Unit 1

Topic 1.3 develops estimates from graphs, and Topic 1.4 develops estimates from tables. Topics 1.5 and 1.6 introduce algebraic properties and manipulation. Later topics cover infinite limits, limits at infinity, continuity and the Intermediate Value Theorem. Unit 2 uses limits to define derivatives.

The practice collection covers the full unit. Use it as a complete unit check after studying the remaining topics.

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