Reading limits off a graph AB & BC

The procedure is mechanical:

1. Put a finger on the curve well to the left of $x = a$ and slide right. Note the height you are heading toward. That is $\lim_{x\to a^-} f(x)$.
2. Repeat from the right, sliding left. That is $\lim_{x\to a^+} f(x)$.
3. If the two heights match, that common value is the limit. If not, the limit does not exist.

Notice that this procedure never requires you to look at $x = a$ itself. Whatever is drawn there, a solid dot, an open circle, or nothing at all, does not enter the calculation.

Graph conventions carry precise information:

1. An open circle at $(a, L)$ says the curve approaches height $L$ but the function does not take that value there. It marks the limit.
2. A solid dot at $(a, k)$ says $f(a) = k$. It marks the function value.
3. Both at the same input is a removable discontinuity: the limit is $L$, the value is $k$, and $L \ne k$.
4. An open circle with no solid dot anywhere on that vertical line means $f(a)$ is undefined, while the limit still exists.

Reporting the solid dot as the limit is the single most common graph-reading error. The solid dot answers "what is $f(a)$", which is a different question.

Where the curve shoots off along a vertical asymptote, the outputs do not approach any real number, so the limit does not exist. Writing $\lim_{x\to a} f(x) = \infty$ is standard shorthand, but it describes how the limit fails, not a value it takes. Infinity is not a number the outputs get close to; it is a statement that they grow without bound.

Read each side separately here too, because the sides frequently disagree in sign. For $\frac{1}{x}$ at 0 the left side falls to $-\infty$ and the right side climbs to $+\infty$. Reporting a single infinite value hides that completely. When both sides run the same way, as with $\frac{1}{x^2}$ at 0, you may write $\lim_{x\to 0}\frac{1}{x^2} = \infty$, still meaning the limit fails in a specific describable way.

A graph gives an estimate, and the estimate is only as good as the grid. If gridlines are one unit apart, reporting a limit of $2.35$ overstates what the picture supports; $2.5$ or "about 2" is the truthful reading. If the exact value matters, the graph tells you which algebraic answer to expect and the algebra confirms it.

Watch also for curves that oscillate faster and faster near a point, such as $\sin\left(\frac{1}{x}\right)$ near 0. There the graph never settles on a height from either side, and the limit fails for a reason that is neither a jump nor an asymptote. A picture that looks like solid ink near the axis is telling you something real.