1.2 Defining Limits and Using Limit Notation

Topic 1.1 found an instantaneous velocity by watching what average velocities approach as the time interval shrinks. That idea of approaching a value, without needing to arrive at it, is the limit. This page gives it a precise meaning and the notation used for it everywhere else in the course.

What a limit means

A limit describes the value that a function approaches as its input approaches a particular number. The notation

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

means that $f(x)$ approaches $L$ as $x$ approaches $a$.

The arrow in $x\to a$ describes the input’s motion; it does not require $x=a$. This distinction matters because the function may be undefined at $a$, or its actual value $f(a)$ may differ from the value approached nearby.

Consider

$$ f(x)=\frac{x^2-4}{x-2}. $$

At $x=2$, the expression is undefined because its denominator is $0$. For $x\ne2$, factor and simplify:

$$ f(x)=\frac{(x-2)(x+2)}{x-2}=x+2. $$

Therefore,

$$ \lim_{x\to2}\frac{x^2-4}{x-2}

\lim_{x\to2}(x+2)

4, $$

even though $f(2)$ does not exist.

Reading a limit from representations

A graph gives a visual limit procedure: locate $x=a$, then inspect the $y$-values the curve approaches as $x$ moves toward $a$ from the left and from the right. If both sides approach the same number $L$, then $\lim_{x\to a}f(x)=L$; if they approach different values, the two-sided limit does not exist.

A removable hole illustrates the distinction clearly: a curve may approach $y=4$ from both directions while displaying an open circle at $(2,4)$ and a filled point elsewhere—or no filled point at all. The limit is still $4$ because the limit concerns nearby behavior, not necessarily the point’s plotted value.

Nearby numerical values can also support a limit estimate. The reliable question is not “What is the value at $x=a$?” but “What values does $f(x)$ approach when $x$ is chosen increasingly close to $a$?” Extended table-based estimation belongs to Topic 1.4.

Reasoning and representation

These topics emphasize CHA 1.1 Introducing Calculus: Can Change Occur at an Instant? and LIM 1.2 Defining Limits and Using Limit Notation, each associated with Mathematical Practice 2—Connecting Representations. A strong conclusion connects a graphical, numerical, analytical, or verbal observation rather than reporting an unsupported number.

The work also uses Mathematical Practice 1—Implementing Mathematical Processes, when algebraic limit properties or simplification are selected; Mathematical Practice 3—Justification, when left- and right-hand behavior is checked; and Mathematical Practice 4—Communication and Notation, when the limit is written with correct symbols, units, and a clear interpretation. When a calculator is used for a numerical estimate, keep appropriate precision during computation and round the final reported value appropriately.

Misconception check — “The limit equals $f(a)$.”
Not always. The limit is the value approached near $a$; $f(a)$ is the function’s actual assigned value at $a$. They are equal only when the function is defined there with that same value.

Retrieval check: If a graph approaches $y=-3$ from both sides as $x\to5$, but the filled point at $x=5$ has height $2$, state $\lim_{x\to5}f(x)$ and $f(5)$. Why are the two answers different?