What a limit says AB & BC

The statement $\lim_{x\to a} f(x) = L$ means: the outputs $f(x)$ can be made as close to $L$ as you like by taking $x$ close enough to $a$, without ever letting $x$ equal $a$.

That exclusion is written into the definition on purpose. Three situations are all perfectly consistent with $\lim_{x\to 2} f(x) = 5$:

1. $f(2) = 5$. The function goes where it was heading.
2. $f(2) = 1$. The graph has a hole at height 5 and a dot at height 1.
3. $f(2)$ is undefined. Nothing is plotted at $x = 2$ at all.

In every case the limit is 5, because in every case the nearby outputs cluster around 5. The limit cannot see the point. Later, "continuous at $a$" is precisely the name for case 1, where the approach and the arrival happen to agree.

Sometimes the two directions of approach behave differently, so the notation separates them:

$$\lim_{x\to a^-} f(x) \quad\text{(from the left)}, \qquad \lim_{x\to a^+} f(x) \quad\text{(from the right)}.$$

Read the superscript as which side the inputs come from: $a^-$ means inputs slightly less than $a$, and $a^+$ means inputs slightly greater. It does not mean the sign of the output.

The existence rule is then a single sentence:

$$\lim_{x\to a} f(x) = L \iff \lim_{x\to a^-} f(x) = L \ \text{ and } \ \lim_{x\to a^+} f(x) = L.$$

Both sides must exist and agree. If they disagree, the two-sided limit does not exist, no matter how well behaved each side is on its own.

Three structures reliably make the sides disagree, and all three are worth recognizing on sight:

1. Piecewise boundaries. With one rule for $x < 2$ and another for $x \ge 2$, each side must be evaluated with its own rule.
2. Absolute value corners. $\frac{|x|}{x}$ is $-1$ for $x<0$ and $+1$ for $x>0$, so at 0 the sides give $-1$ and $1$ and no limit exists.
3. Vertical asymptotes. The two sides often run to opposite infinities.

The characteristic error is evaluating a two-sided limit at a join using only one branch, usually whichever rule is written first or happens to include the endpoint. Which branch owns the equality sign is irrelevant to the limit: the limit never uses the value at the point, so a piecewise definition at $x = a$ has no bearing on it.

A few conventions save marks. The variable and the target both matter: $\lim_{x\to 3}$ and $\lim_{t\to 3}$ describe different things if the expression contains both letters. Write $\lim$ with its subscript every time until the limit is actually taken; an equals sign between an expression and its limit value, with no limit symbol, is a different and usually false claim.

When a limit fails to exist, say so rather than writing a value. "DNE" is a complete answer. Writing $\lim_{x\to 0}\frac{|x|}{x} = \pm 1$ is not a statement about anything; the two one-sided limits are $-1$ and $1$, and the two-sided limit simply does not exist.