Limit Laws, calculus lesson

Proving every limit straight from the epsilon-delta definition would be slow. The limit laws let you split a complicated expression into pieces whose limits you already know and put the answers back together. Each law comes with conditions, and most mistakes with limits come from ignoring them.

Key terms

Sum law

Definition Let $a$ be a real number and suppose

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

both exist as real numbers. Then

\lim_{x \to a} \bigl(f(x) + g(x)\bigr) = L + M.

These assumptions stay in force for every law on this page, and every law also holds for one-sided limits.

In plain words The limit of a sum is the sum of the limits, as long as both limits exist.

Example Since $\lim_{x \to 2} x^2 = 4$ and $\lim_{x \to 2} 3x = 6$ , the sum law gives

\lim_{x \to 2} (x^2 + 3x) = 4 + 6 = 10.

Difference law

Definition Under the same assumptions,

\lim_{x \to a} \bigl(f(x) - g(x)\bigr) = L - M.

In plain words The limit of a difference is the difference of the limits.

Example Near $x = 1$ the pieces tend to $x^3 \to 1$ and $4x \to 4$ , so

\lim_{x \to 1} (x^3 - 4x) = 1 - 4 = -3.

Constant multiple law

Definition For any constant $c$ ,

\lim_{x \to a} c f(x) = cL.

In plain words Constant factors can be pulled out in front of the limit.

Example Pulling out the factor 5 gives

\lim_{x \to -1} 5x^2 = 5 \lim_{x \to -1} x^2 = 5 \cdot 1 = 5.

Product law

Definition Under the assumptions above,

\lim_{x \to a} f(x) g(x) = LM.

In plain words The limit of a product is the product of the limits, provided both limits exist. If one factor has no limit, this law tells you nothing, and the product may or may not have a limit.

Example The factors tend to $3$ and $1$ as $x \to 2$ , so

\lim_{x \to 2} (x + 1)(x^2 - 3) = 3 \cdot 1 = 3.

On the other hand $x \sin (1/x)$ has limit $0$ at $0$ even though $\sin (1/x)$ has no limit there. The product law cannot show that, and a different theorem is needed.

Quotient law with a nonzero limiting denominator

Definition If in addition $M \neq 0$ , then

\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{L}{M}.

When $M \neq 0$ , the function $g$ is automatically nonzero for all $x \neq a$ close enough to $a$ , so the quotient is defined there. When $M = 0$ the law does not apply, and the limit may exist or not.

In plain words Divide the limits, but only when the bottom limit is not zero.

Example The denominator tends to $2 \neq 0$ , so

\lim_{x \to 3} \frac{x^2 + 1}{x - 1} = \frac{10}{2} = 5.

Compare two cases with $M = 0$ . For $(x^2 - 9)/(x - 3)$ at $3$ the limit is $6$ after cancelling, while $1/x$ at $0$ has no limit at all. The law gives neither answer.

Power law

Definition For every positive integer $n$ ,

\lim_{x \to a} \bigl(f(x)\bigr)^n = L^n.

This follows from the product law applied $n - 1$ times. Together with $\lim_{x \to a} x = a$ it gives $\lim_{x \to a} x^n = a^n$ .

In plain words Raise the limit to the same power.

Example The base $3x - 4$ tends to $2$ as $x \to 2$ , so

\lim_{x \to 2} (3x - 4)^5 = 2^5 = 32.

Root law and real-domain conditions

Definition Let $n$ be a positive integer. If $n$ is odd, then

\lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{L}.

If $n$ is even, the same holds provided $f(x) \ge 0$ for all $x \neq a$ in some open interval around $a$ , so that the root is a real number there. That condition forces $L \ge 0$ . It holds automatically when $L > 0$ , because then $f(x)$ is close to the positive number $L$ , and so positive, for $x$ near $a$ . When $L = 0$ it has to be checked.

In plain words Take the root of the limit. For square roots and other even roots, make sure you are never taking the root of a negative number near $a$ .

Example The insides tend to $25 > 0$ and $8$ , so

\lim_{x \to 4} \sqrt{x^2 + 9} = \sqrt{25} = 5 \quad \text{and} \quad \lim_{x \to -3} \sqrt[3]{x^2 - 1} = \sqrt[3]{8} = 2.

By contrast, $\sqrt{-x^2}$ is a real number only at $x = 0$ , so it has no limit at $0$ in the real numbers, even though $-x^2 \to 0$ .

Proof of a limit law

Definition Here is the proof of the sum law. Let $\varepsilon > 0$ . Since $f(x) \to L$ , there is $\delta_1 > 0$ with $\lvert f(x) - L \rvert < \varepsilon/2$ whenever $0 < \lvert x - a \rvert < \delta_1$ . Since $g(x) \to M$ , there is $\delta_2 > 0$ with $\lvert g(x) - M \rvert < \varepsilon/2$ whenever $0 < \lvert x - a \rvert < \delta_2$ . Let $\delta = \min(\delta_1, \delta_2)$ . If $0 < \lvert x - a \rvert < \delta$ , the triangle inequality gives

\lvert (f(x) + g(x)) - (L + M) \rvert \le \lvert f(x) - L \rvert + \lvert g(x) - M \rvert < \frac{\varepsilon}{2} + \frac{\varepsilon}{2} = \varepsilon.

In plain words Split the error budget in half, give each function its share, and use the smaller of the two $\delta$ values so both promises hold at once.

Example The constant multiple law has a similar proof. If $c = 0$ both sides are $0$ . If $c \neq 0$ , let $\varepsilon > 0$ and pick $\delta > 0$ so that $\lvert f(x) - L \rvert < \varepsilon / \lvert c \rvert$ whenever $0 < \lvert x - a \rvert < \delta$ . Then for those $x$

\lvert c f(x) - cL \rvert = \lvert c \rvert \, \lvert f(x) - L \rvert < \lvert c \rvert \cdot \frac{\varepsilon}{\lvert c \rvert} = \varepsilon.

Check yourself

Suppose $\lim_{x \to a} f(x) = 4$ and $\lim_{x \to a} g(x) = -2$ . Find the limit

\lim_{x \to a} \frac{3f(x) - g(x)^2}{f(x) + g(x)}.

Show the answer

The numerator tends to $12 - 4 = 8$ and the denominator to $2 \neq 0$ , so the limit is $8/2 = 4$ .

Find $\lim_{x \to 1} \sqrt[3]{x^3 + 7x}$ .

Show the answer

The inside tends to $1 + 7 = 8$ , and the cube root law needs no sign condition, so the limit is $\sqrt[3]{8} = 2$ .

Why can the quotient law not be used directly for the limit below, and what is the limit?

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

Show the answer

The denominator tends to $0$ , so the condition $M \neq 0$ fails. For $x \neq 1$ the quotient equals $x + 1$ , so the limit is $2$ .

Limit Laws

Theory, part 13

Practice, part 14

Key terms

Sum law

Difference law

Constant multiple law

Product law

Quotient law with a nonzero limiting denominator

Power law

Root law and real-domain conditions

Proof of a limit law

Check yourself

How this lesson connects

Builds on

Used later in