Limit Laws and the ε–δ Definition

The intuitive definition explains what a limit means but not how to compute one without a table. The Limit Laws close that gap: limits respect the arithmetic operations, so a complicated limit assembles from trivial ones. Where the laws stall, on the indeterminate $0/0$ form, algebra or the Squeeze Theorem takes over, and the $\epsilon$-$\delta$ definition replaces arbitrarily close with quantified inequalities that support proof.

The Limit Laws

Stated in words: the limit of a sum is the sum of the limits, the limit of a product is the product of the limits, and so on, with the one caveat that a quotient needs a nonzero denominator limit.¹ Repeated application of the Product Law to $f=g$ gives a Power Law, and inverting a power gives a Root Law:

the root form requiring $\lim f(x)>0$ when $n$ is even. Two base cases seed everything, and both are immediate from the definition:

From the base cases and the laws, a limit like ${\lim }_{x\to 2}(x^3+2x^2-1)$ evaluates by structure alone: powers of the limit $2$, scaled and summed, giving $8+8-1=15$. Written out, each step names the law that licenses it.

For a rational function the Quotient Law applies once the denominator's limit is seen to be nonzero.

Direct substitution

That last computation is an instance of a general shortcut.

For these functions, evaluating the limit is nothing more than plugging in $a$. This property characterizes the continuous functions. Substitution disposes of any limit where the direct value is defined; the remaining cases are those where it is not.

Indeterminate forms

When substitution yields $0/0$, the expression is an indeterminate form: the quotient's limit could be any number, and the form alone decides nothing. The value $0$ in the denominator flags a common factor of $x-a$ shared with the numerator, and removing it usually reveals the limit.

Factoring. For

substitution gives $0/0$. Factor and cancel the offending $x-1$, legitimate because the limit ignores $x=1$ itself:

Rationalizing. When a square root blocks factoring, multiply by the conjugate. For

multiply top and bottom by $\sqrt{t^2+9}+3$:

Now substitution works, giving $\frac{1}{\sqrt{9}+3}=\frac{1}{6}$.

Combining fractions. A quotient of fractions clears once the numerator is put over a common denominator.

The three tactics — factor-and-cancel, rationalize, and combine fractions — handle most algebraic $0/0$ forms.

The Squeeze Theorem

Some limits resist algebra entirely, and for those a comparison argument works. First, limits preserve the ordering of functions.

Trapping a function between two others that share a limit pins its limit as well.

The standard application handles an oscillation that never settles.

The oscillation is real, but its amplitude is crushed to zero by the $x^2$ envelope.

The precise definition

Arbitrarily close and sufficiently close carry the intuitive definition but are too vague to prove theorems with. The $\epsilon$-$\delta$ definition replaces them with quantified inequalities.

Here $\epsilon$ measures a target tolerance on the output and $\delta$ a tolerance on the input. The statement reads geometrically: for any horizontal band of half-width $\epsilon$ around $y=L$, there is a vertical strip of half-width $\delta$ around $x=a$ whose graph (excluding $x=a$) lands entirely inside the band. The clause $0<|x-a|$ excludes $x=a$, matching the limit's indifference to $f(a)$.

The definition frames a contest between two parties. One names a tolerance $\epsilon$, however small; the other must supply a $\delta$ that keeps the output within $\epsilon$. The limit equals $L$ precisely when the second party can always answer, no matter how small the first makes $\epsilon$.

A worked ε–δ proof

To prove $\underset{x\to 3}{\lim }(4x-5)=7$, work backward from the target to find the $\delta$, then present the argument forward.

Finding $\delta$. The output tolerance is $|(4x-5)-7|<\epsilon$. Simplify the left side:

This is below $\epsilon$ exactly when $|x-3|<\epsilon /4$, so $\delta =\epsilon /4$ is the choice.

The structure is the same for every such proof: solve the output inequality for $|x-a|$ to discover $\delta$, then verify.

A nonlinear function makes the last step harder. The factor multiplying $|x-a|$ is no longer constant, so it must first be bounded by restricting $x$ to a small window, which forces $\delta$ to be the smaller of two constraints.

A one-sided limit uses the same template with the input interval halved to one side. The right-hand definition asks that $|f(x)-L|<\epsilon$ whenever $a<x<a+\delta$.

The Limit Laws themselves rest on this definition: the Sum Law, for instance, follows by choosing $\delta$ so that $|f(x)-L|$ and $|g(x)-M|$ are each below $\epsilon /2$, whence $|(f+g)(x)-(L+M)|<\epsilon$ by the triangle inequality. Once proved, the laws evaluate complicated limits without returning to $\epsilon$ and $\delta$.