Limits — Definition, Rules & How to Evaluate

In calculus, a limit is the value a function approaches as the input approaches a point. You use limits when direct substitution is unhelpful, especially near holes, jumps, or expressions that produce $0/0$.

In symbols,

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

means that when $x$ moves near $a$, the values of $f(x)$ move near $L$.

The key point is that a limit cares about nearby behavior, not only the exact value at $x=a$. The function could equal a different number there, or be undefined there, and the limit could still exist.

Limit definition: approach, not arrival

The word "limit" is about approach, not arrival. If

$$\lim_{x \to 2} f(x) = 5$$

that does not automatically mean $f(2) = 5$. It means $f(x)$ gets close to $5$ when $x$ gets close to $2$ from both sides.

This is why limits matter for piecewise functions, rational expressions, and graphs with holes. They let you describe what the function is doing near a point even when the point itself is problematic.

Limit laws you can use safely

When the simpler limits exist, you can combine them to evaluate more complicated limits.

If

$$\lim_{x \to a} f(x) = L \qquad \text{and} \qquad \lim_{x \to a} g(x) = M,$$

then:

$$\lim_{x \to a} \left(f(x) + g(x)\right) = L + M$$ $$\lim_{x \to a} \left(c f(x)\right) = cL$$ $$\lim_{x \to a} \left(f(x)g(x)\right) = LM$$ $$\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{L}{M} \qquad \text{if } M \ne 0$$

The condition $M \ne 0$ matters. If the denominator limit is $0$, the quotient law does not justify the step.

For polynomials and many familiar functions, direct substitution works because the function is continuous at the point you are checking.

How to evaluate a basic limit

Most basic limit problems follow the same order:

1. Try direct substitution.
2. If you get an ordinary real number, that is the limit.
3. If you get an indeterminate form such as $0/0$, simplify first.
4. If the expression may behave differently on the two sides, compare one-sided limits.

One-sided notation looks like this:

$$\lim_{x \to a^-} f(x) \qquad \text{and} \qquad \lim_{x \to a^+} f(x)$$

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

Worked example: a $0/0$ limit

Evaluate

$$\lim_{x \to 1} \frac{x^2 - 1}{x - 1}$$

Direct substitution gives

$$\frac{1^2 - 1}{1 - 1} = \frac{0}{0}$$

That is not the answer. It only tells you direct substitution did not finish the problem.

Factor the numerator:

$$x^2 - 1 = (x - 1)(x + 1)$$

For $x \ne 1$,

$$\frac{x^2 - 1}{x - 1} = \frac{(x - 1)(x + 1)}{x - 1} = x + 1$$

Now the limit is easier:

$$\lim_{x \to 1} (x + 1) = 2$$

So

$$\lim_{x \to 1} \frac{x^2 - 1}{x - 1} = 2$$

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

Common mistakes when evaluating limits

1. Treating $0/0$ as a final value. It is a warning sign, not a solution.
2. Assuming the limit must equal $f(a)$. That only happens when the function is continuous at $a$.
3. Using the quotient law when the denominator limit is $0$. That condition breaks the rule.
4. Ignoring left-hand and right-hand behavior. If the two sides approach different values, the limit does not exist.
5. Canceling factors without noting the condition. In the worked example, canceling $x-1$ is valid only for $x \ne 1$, which is enough for the limit because limits use nearby points.

Where limits are used in calculus

Limits are the starting point for several core ideas in calculus. They are used to

1. define derivatives,
2. describe continuity,
3. analyze behavior near asymptotes or endpoints, and
4. justify simplifications near points where a formula is not directly defined.

If you go on to derivatives, integrals, or infinite sequences and series, limits are part of the language behind all of them.

A quick check before you move on

After you solve a limit, ask one question: do nearby values really head toward your answer from both sides?

That quick check catches many mistakes, especially in piecewise functions and rational expressions.

Try a similar limit

Try

$$\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$$

Use the same pattern: substitute, notice the $0/0$, factor, simplify, and substitute again. If you want the next step, try your own version with a piecewise function and check whether the left-hand and right-hand limits match.