AP Calculus AB: Limits and Continuity

Limits and continuity are the language of calculus. They formalize what it means for a function to “approach” a value and when a function behaves predictably without breaks or jumps. In AP Calculus AB, these ideas sit at the foundation of derivatives and integrals, so learning them well pays off throughout the course.

What a Limit Means

A limit describes the value a function $f(x)$ approaches as $x$ gets close to some number $a$. We write this as:

${\lim }_{x\to a}f(x)=L$

This statement is about behavior near $a$, not necessarily at $a$. In particular, $f(a)$ might be undefined, or it might be defined but different from $L$. The limit only cares about what happens as $x$ approaches $a$.

One-Sided Limits

Sometimes behavior differs depending on whether you approach from the left or the right.

• Left-hand limit: ${\lim }_{x\to a^{-}}f(x)$
• Right-hand limit: ${\lim }_{x\to a^{+}}f(x)$

A two-sided limit exists exactly when both one-sided limits exist and are equal:

If ${\lim }_{x\to a^{-}}f(x)={\lim }_{x\to a^{+}}f(x)=L$, then ${\lim }_{x\to a}f(x)=L$.

Limits at Infinity

Limits also describe end behavior.

• ${\lim }_{x\to \infty }f(x)=L$ means $f(x)$ approaches $L$ as $x$ grows without bound.
• ${\lim }_{x\to -\infty }f(x)=L$ means the same as $x$ becomes very negative.

These are central when discussing horizontal asymptotes and long-run behavior.

Techniques and Limit Laws

For many functions, limits can be computed using basic algebra and the limit laws, which let you “push the limit through” common operations when the needed limits exist:

• Sum: $\lim (f+g)=\lim f+\lim g$
• Product: $\lim (fg)=(\lim f)(\lim g)$
• Quotient: $\lim \frac{f}{g}=\frac{\lim f}{\lim g}$, provided the denominator limit is not zero
• Constant multiple: $\lim (cf)=c\lim f$
• Power and root rules for appropriate domains

Direct Substitution and Indeterminate Forms

If $f$ is a polynomial or a rational function (with nonzero denominator at $a$), direct substitution usually works.

The classic complication is an indeterminate form like $\frac{0}{0}$. That does not mean the limit is zero. It means more work is needed, often by factoring and simplifying.

Example (typical AP pattern): if you get $\frac{x^2-9}{x-3}$ as $x\to 3$, factoring gives $\frac{(x-3)(x+3)}{x-3}=x+3$ for $x\ne 3$, so the limit is $6$.

This illustrates an important theme: a limit can exist even when the function is not defined at the point.

The Squeeze Theorem

The Squeeze Theorem is used when a function is trapped between two simpler functions that share the same limit.

If $g(x)\le f(x)\le h(x)$ near $a$ (not necessarily at $a$), and

${\lim }_{x\to a}g(x)={\lim }_{x\to a}h(x)=L$,

then ${\lim }_{x\to a}f(x)=L$.

A common AP Calculus AB example is $f(x)=x^2\sin (1/x)$ as $x\to 0$. Since $-1\le \sin (1/x)\le 1$, multiplying by $x^2$ gives $-x^2\le x^2\sin (1/x)\le x^2$, and both outer limits go to $0$, so the squeezed function’s limit is also $0$.

The power of the theorem is that it avoids needing to tame oscillation directly.

The Epsilon-Delta Definition (Why Limits Are Rigorous)

While many limits are computed with algebra, calculus demands a precise definition. The epsilon-delta definition formalizes what it means for ${\lim }_{x\to a}f(x)=L$:

For every $\epsilon >0$, there exists a $\delta >0$ such that if $0<|x-a|<\delta$, then $|f(x)-L|<\epsilon$.

Interpretation:

• $\epsilon$ is how close you want $f(x)$ to be to $L$ (vertical closeness).
• $\delta$ is how close $x$ must be to $a$ to guarantee that (horizontal closeness).

This definition is central on AP exams when reasoning about why a limit exists, or when connecting limit statements to function behavior. You may not be asked to construct full proofs often, but you should understand the logic: controlling input closeness controls output closeness.

Continuity: The “No Surprises” Condition

A function $f$ is continuous at $x=a$ if all three conditions hold:

1. $f(a)$ is defined.
2. ${\lim }_{x\to a}f(x)$ exists.
3. ${\lim }_{x\to a}f(x)=f(a)$.

In plain terms, the function value matches the value the function approaches.

Continuity on an Interval

• Continuous on $(a,b)$: continuous at every point inside.
• Continuous on $[a,b]$: continuous on $(a,b)$ and has appropriate one-sided continuity at endpoints.

Polynomials are continuous everywhere. Rational functions are continuous wherever their denominators are not zero. Many AP problems rely on recognizing these facts quickly.

Types of Discontinuities

Discontinuities describe how continuity fails. In AP Calculus AB, three common categories appear.

Removable Discontinuity (A “Hole”)

The limit exists, but the function is missing the point or has the “wrong” value.

• Example behavior: ${\lim }_{x\to a}f(x)=L$ exists, but $f(a)$ is undefined, or $f(a)\ne L$.
• Often comes from factors that cancel, creating a hole in a graph.

A removable discontinuity can be fixed by redefining $f(a)=L$.

Jump Discontinuity

The left and right limits exist but are different:

${\lim }_{x\to a^{-}}f(x)\ne {\lim }_{x\to a^{+}}f(x)$

This is typical for piecewise functions where the pieces meet at different heights.

Infinite Discontinuity (Vertical Asymptote)

The function grows without bound near $a$, such as:

${\lim }_{x\to a}f(x)=\infty$ or $-\infty$

Rational functions commonly show this behavior when the denominator approaches zero but does not cancel with the numerator.

The Intermediate Value Theorem (IVT)

The Intermediate Value Theorem is one of the most useful consequences of continuity. It states:

If $f$ is continuous on $[a,b]$ and $N$ is any number between $f(a)$ and $f(b)$, then there exists at least one $c$ in $[a,b]$ such that $f(c)=N$.

Practical meaning: a continuous function cannot “skip” values. This is why continuity is so important in root-finding and modeling.

How IVT Is Used on AP Calculus AB

A typical application is proving that an equation has a solution on an interval without solving it exactly. For example, to show a function has a root, you check that:

• $f$ is continuous on $[a,b]$
• $f(a)$ and $f(b)$ have opposite signs

Since $0$ lies between $f(a)$ and $f(b)$, IVT guarantees some $c$ with $f(c)=0$.

This does not tell you how many solutions exist or where they are precisely, only that at least one exists.

How Limits and Continuity Connect to the Rest of Calculus

Limits define derivatives: the derivative at a point is a limit of average rates of change. Continuity supports theorems and methods used later, including guaranteed root existence (IVT) and the predictable behavior required for many applications. If you can compute limits reliably, classify discontinuities accurately, and use continuity theorems with confidence, you have the groundwork needed for everything that follows in AP Calculus AB.