Limits — Definition, Rules & How to Evaluate

In calculus, a limit is the value a function approaches as the input approaches a point. The defining notation,

means that as $x$ moves near $a$, the values of $f(x)$ move near $L$. The crucial idea is approach, not arrival: a limit cares about nearby behavior, not the exact value at $x = a$. The function can equal a different number there, or be undefined there, and the limit can still exist.

When you need limits

You reach for limits when direct substitution is unhelpful — near holes, jumps, or expressions that produce $0/0$. If

that does not force $f(2) = 5$; it only says $f(x)$ gets close to $5$ as $x$ gets close to $2$ from both sides. This is exactly why limits matter for piecewise functions, rational expressions, and graphs with holes: they describe what a function does near a troublesome point.

When the function is continuous at the point, direct substitution simply works — that covers polynomials and many familiar functions.

The procedure, step by step

1. Try direct substitution. Replace $x$ with $a$ and see what you get.
2. Accept an ordinary number. If substitution yields a real number, that is the limit.
3. Simplify an indeterminate form. If you get $0/0$, factor or rewrite, then substitute again.
4. Compare one-sided limits when needed. If the function may behave differently on each side, check both.

One-sided notation looks like

and the full limit exists only when both one-sided limits exist and are equal.

When simpler limits exist, the limit laws let you combine them. If $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = M$, then

The condition $M \ne 0$ matters: when the denominator limit is $0$, the quotient law no longer justifies the step.

Full worked example: a $0/0$ limit

Evaluate

Direct substitution gives

which is a warning sign, not an answer. Factor the numerator:

For $x \ne 1$,

so

The function is undefined at $x = 1$, yet the limit exists because nearby values approach $2$. That is the standard pattern for a removable discontinuity.

Where each step traps people, plus a self-check

Step one traps those who treat $0/0$ as a final value — it is only a flag. Step two traps those who assume the limit must equal $f(a)$, which holds only under continuity. The quotient law traps people who apply it when the denominator limit is $0$. The one-sided step traps anyone who ignores left- and right-hand behavior: if the two sides head to different values, the limit does not exist. And canceling factors is valid only with its condition — in the example, canceling $x - 1$ is legitimate only for $x \ne 1$, which is enough because limits use nearby points.

After any limit, run one self-check: do nearby values really head toward your answer from both sides? That single question catches most mistakes in piecewise functions and rational expressions.

For your own pass, try

using the same rhythm — substitute, notice the $0/0$, factor, simplify, substitute again. Limits are the foundation beneath derivatives, continuity, asymptotic behavior, and the justifications behind simplifications where a formula is not directly defined, so the habit pays off well past this one problem.