IB AA: Limits and Continuity

The concepts of limit and continuity are the bedrock upon which all of calculus is built. Without a firm grasp of limits, the definitions of the derivative and the integral become meaningless. In IB Analysis & Approaches (AA), you move beyond mere calculation to a more formal understanding of these ideas, which is essential for handling the rigorous mathematical arguments you'll encounter in higher-level calculus and analysis.

What a Limit Represents

Intuitively, the limit of a function $f(x)$ as $x$ approaches a value $a$ is the number $L$ that $f(x)$ gets arbitrarily close to as $x$ gets arbitrarily close to $a$, without necessarily ever reaching $a$ or having $f(a)$ defined. Consider the function $f(x)=\frac{\sin x}{x}$. It is undefined at $x=0$, but you can evaluate it for values very close to 0. As you try $x=0.1,0.01,0.001$, you'll find $f(x)$ gets closer and closer to 1. We write this as: $$\underset{x\to 0}{\lim }\frac{\sin x}{x}=1$$

This leads to the idea of one-sided limits. The limit from the right, denoted ${\lim }_{x\to a^{+}}f(x)$, considers only values of $x$ greater than $a$. The limit from the left, ${\lim }_{x\to a^{-}}f(x)$, considers only values less than $a$. For the overall (two-sided) limit to exist and equal $L$, both one-sided limits must exist and be equal to that same $L$.

We also consider limits at infinity. The statement ${\lim }_{x\to \infty }f(x)=L$ means that as $x$ grows larger and larger without bound, the function values $f(x)$ approach the finite number $L$. For example, for $f(x)=1+\frac{1}{x}$, as $x$ becomes enormous, $\frac{1}{x}$ becomes negligible, so the limit is 1.

The Formal "ε-δ" Definition

The IB AA course introduces the formal, epsilon-delta (ε-δ) definition of a limit to solidify the intuitive idea. It states: ${\lim }_{x\to a}f(x)=L$ if, for every number $ϵ>0$ (no matter how small), there exists a number $\delta >0$ such that if $0<|x-a|<\delta$, then $|f(x)-L|<ϵ$.

In plain language, this guarantees you can make $f(x)$ as close as you want to $L$ (within any $ϵ$ distance) by choosing $x$ sufficiently close to $a$ (within some $\delta$ distance). This definition removes any ambiguity about phrases like "approaches" or "gets close to" and is the foundation for proving all subsequent limit theorems.

Limit Laws and Tools

You don't always need the formal definition to evaluate limits. Limit laws provide algebraic shortcuts, assuming the individual limits exist. Key laws include:

• The limit of a sum/difference is the sum/difference of the limits.
• The limit of a product is the product of the limits.
• The limit of a quotient is the quotient of the limits (provided the denominator's limit is not zero).
• ${\lim }_{x\to a}[c\cdot f(x)]=c\cdot {\lim }_{x\to a}f(x)$, where $c$ is a constant.

For indeterminate forms like $\frac{0}{0}$ or $\frac{\infty }{\infty }$, algebraic manipulation (e.g., factoring, rationalizing) is required. A powerful tool for such forms, especially those involving trigonometric functions, is the Squeeze Theorem (or Sandwich Theorem). It states: If $g(x)\le f(x)\le h(x)$ for all $x$ near $a$ (except possibly at $a$), and ${\lim }_{x\to a}g(x)={\lim }_{x\to a}h(x)=L$, then ${\lim }_{x\to a}f(x)=L$. This is how the limit ${\lim }_{x\to 0}\frac{\sin x}{x}=1$ is formally proven.

Continuity: The Bridge Between Points

A function is continuous at a point $x=a$ if three conditions are met:

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

Informally, you can draw the graph of a continuous function at a point without lifting your pen. If a function is continuous at every point in an interval, it is continuous on an interval. Polynomials, rational functions (where defined), and basic trigonometric functions are continuous on their domains.

Discontinuities are breaks in the graph. A removable discontinuity (or hole) occurs when ${\lim }_{x\to a}f(x)$ exists but is not equal to $f(a)$ (or $f(a)$ is undefined). A jump discontinuity occurs when the left-hand and right-hand limits exist but are not equal. An infinite discontinuity occurs when the function values increase or decrease without bound as $x$ approaches $a$.

Foundational Theorems

Two major theorems arise directly from the properties of continuity. The Intermediate Value Theorem (IVT) is both intuitive and powerful. It states: If a function $f$ is continuous on the closed interval $[a,b]$, and $N$ is any number between $f(a)$ and $f(b)$, then there exists at least one number $c$ in $(a,b)$ such that $f(c)=N$.

This theorem guarantees the existence of roots. If $f(a)$ is negative and $f(b)$ is positive (or vice-versa), then by the IVT with $N=0$, there must be some $c$ where $f(c)=0$. It's used not just to find roots, but to prove that a function takes on every intermediate value on an interval.

Common Pitfalls

1. Assuming the limit is just the function value. The limit as $x$ approaches $a$ is concerned with the behavior of $f(x)$ near $a$, not at $a$. A function can have a perfectly defined limit at a point where it is not defined. Always check the function's behavior from both sides.
2. Misapplying limit laws to indeterminate forms. You cannot apply the quotient law if the limit of the denominator is zero. For example, ${\lim }_{x\to 2}\frac{x^2-4}{x-2}$ is of the form $\frac{0}{0}$. The correct approach is to simplify the expression to $x+2$ (for $x\ne 2$) and then take the limit.
3. Confusing continuity with having no "sharp corners." Continuity is about having no breaks or jumps. A function can be continuous yet not differentiable (like $f(x)=|x|$ at $x=0$). The "sharp corner" affects the derivative, not the continuity.
4. Misusing the Intermediate Value Theorem. The IVT requires the function to be continuous on a closed interval. Applying it to a discontinuous function or an open interval can lead to false conclusions. Always verify the continuity condition first.

Summary

• The limit describes the intended height a function approaches as the input nears a value, which is foundational for defining instantaneous rates of change (derivatives).
• Continuity at a point requires the function to be defined, have a limit, and for that limit to equal the function value at that point.
• The Intermediate Value Theorem guarantees that a continuous function on a closed interval takes on every value between its endpoints, a key tool for proving root existence.
• The Squeeze Theorem allows you to find limits of complex functions by "trapping" them between simpler functions with known, equal limits.
• Mastering both the intuitive meaning and the formal epsilon-delta definition of a limit is crucial for progressing to the rigorous calculus demanded by the IB AA HL curriculum and beyond.