Topic 1.2 Defining Limits and Using Limit Notation: express the limit of a function using correct notation, including one-sided limits, and interpret what a limit says about the behavior of a function near a point.

What this topic is asking

The College Board (Topic 1.2) wants you to state, in correct notation, what a limit is: the value a function's output approaches as the input approaches some number, regardless of the function's value (or lack of one) at that number. You must distinguish two-sided from one-sided limits and know when a limit does not exist.

Defining a limit

This is why a limit can exist where the function has a hole. For $g(x) = \frac{x^2 - 4}{x - 2}$, the value $g(2)$ is undefined, yet for every $x \neq 2$ we have $g(x) = x + 2$, so $\lim_{x \to 2} g(x) = 4$. The function never reaches the input $2$, but it approaches the output $4$.

One-sided limits

The arrow can carry a sign that fixes the direction of approach:

• $\lim_{x \to a^-} f(x)$ is the left-hand limit: $x$ approaches $a$ through values less than $a$.
• $\lim_{x \to a^+} f(x)$ is the right-hand limit: $x$ approaches $a$ through values greater than $a$.

When a limit does not exist

A two-sided limit can fail to exist for three common reasons:

• The one-sided limits disagree (a jump), as in a step or piecewise function.
• The function grows without bound near $a$ (an infinite limit), for example $\frac{1}{x^2}$ as $x \to 0$.
• The function oscillates and never settles, for example $\sin\left(\frac{1}{x}\right)$ as $x \to 0$.

In each case there is no single finite value the outputs approach from both sides.

The limit describes a trend, not a destination reached

The deepest idea in this topic is that a limit is about the approach, not arrival. The phrasing "$f(x)$ gets arbitrarily close to $L$" means that you can make $f(x)$ as near to $L$ as you like by taking $x$ close enough to $a$ - it does not require $f(x)$ ever to equal $L$, nor $x$ ever to equal $a$. This is why a function can have a limit at a point it never actually reaches, and why redefining or even deleting the single value $f(a)$ has no effect on $\lim_{x \to a} f(x)$. Holding this "trend, not destination" picture in mind prevents the most common conceptual error in the unit, which is to compute $f(a)$ and call it the limit. For continuous functions the two happen to coincide, but the limit concept is built precisely to handle the cases where they do not, such as holes and removable discontinuities you will meet later in the unit.

Notation discipline

Write limits fully and correctly, because the AP exam awards notation. Always include the variable and the value it approaches: $\lim_{x \to a}$, not a bare "lim". Use the superscript minus or plus only when you mean a one-sided limit. When a limit is infinite, you may write $\lim_{x \to a} f(x) = \infty$ to describe the behavior, but remember that this still means the (finite) limit does not exist. Sloppy notation - omitting the arrow, writing "$\lim f = 5$" with no variable, or mixing up the one-sided superscripts - is penalized on free-response questions even when the reasoning is sound, so it is worth making correct notation automatic from the very first topic. The clearer your notation, the easier it is for a marker to award every available point.