Approaching a Limit Numerically and Graphically

Definition For a number $a$ and a radius $\delta > 0$, the inputs within $\delta$ of $a$ other than $a$ itself are the numbers $x$ with

which form the open interval from $a - \delta$ to $a + \delta$ with $a$ removed. Let $f$ be defined at all of these inputs for some $\delta$. We say that $f(x)$ approaches the number $L$ as $x$ approaches $a$, and write

if the outputs can be made as close to $L$ as we like by taking $x$ close enough to $a$ with $x \ne a$. Precisely, for every $\varepsilon > 0$ there is a $\delta > 0$ such that every input with $0 < \lvert x - a \rvert < \delta$ gives $\lvert f(x) - L \rvert < \varepsilon$. So the limit depends only on the values of $f$ at nearby inputs, never on $f(a)$ and never on values far from $a$.

Definition The function value $f(a)$ is the output at the single input $a$, when $a$ is in the domain. The limiting value is the limit of $f(x)$ as $x$ approaches $a$, when that limit exists. Either one can exist without the other, and when both exist they can differ. When both exist and are equal, the graph has no break at $a$, and calculus calls $f$ continuous at $a$.

Definition The left-hand limit

means that $f(x)$ approaches $L$ as $x$ approaches $a$ through inputs $x < a$ only. The right-hand limit, written with $x \to a^+$, uses inputs $x > a$ only. Each one needs $f$ to be defined on an open interval just to that side of $a$. The two-sided limit exists and equals $L$ exactly when both one-sided limits exist and both equal $L$.

Definition A numerical table for a limit at $a$ lists $f(x)$ at inputs that approach $a$ from both sides, such as $a \pm 0.1$, $a \pm 0.01$ and $a \pm 0.001$, and never at $a$ itself. Outputs that close in on one number from both sides suggest the limit. A calculator or computer rounds every number to a fixed count of digits, so at inputs extremely close to $a$, subtracting two nearly equal numbers can wipe out the correct digits.

Definition A graph window, also called a viewing window, is the rectangle of the plane that a graphing tool shows, with $x$ from $x_{\min}$ to $x_{\max}$ and $y$ from $y_{\min}$ to $y_{\max}$. The tool computes $f$ at finitely many sample inputs and joins the dots. So the picture can hide a hole or a single moved point, it can draw a false steep segment across a vertical asymptote, and it can turn fast oscillation into a pattern that is not there. A window centered at $(a, L)$ with half-width $\delta$ and half-height $\varepsilon$ pictures the limit condition. If every point of the graph with $0 < \lvert x - a \rvert < \delta$ lies strictly between the top and bottom edges, then those inputs give $\lvert f(x) - L \rvert < \varepsilon$.

Definition The limit of $f$ at $a$ does not exist when no single real number $L$ meets the definition. There are three common ways this happens. In a jump, both one-sided limits exist but they differ. In unbounded behavior, the outputs on at least one side grow past every bound in size. In oscillation, there are two numbers $p < q$ such that every interval around $a$, however small, contains inputs $x \ne a$ with $f(x) \le p$ and inputs $x \ne a$ with $f(x) \ge q$. When the outputs grow without bound through positive values on both sides we write $\lim_{x \to a} f(x) = \infty$, but this only describes how the limit fails, since $\infty$ is not a real number. A two-sided limit also needs $f$ to be defined at nearby inputs on both sides, so at an endpoint of the domain only a one-sided limit can make sense.

Definition A table checks only finitely many inputs, and a graph shows only what its window and its sampling allow. Both give evidence and a good guess, never a proof. A proof is an argument that covers every input in some interval around $a$ with $a$ removed, usually algebra that rewrites $f(x)$ for $x \ne a$ in a form whose behavior near $a$ is clear, or an inequality that bounds $\lvert f(x) - L \rvert$.