Limit Laws

Suppose

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

For a constant $c$ and positive integer $n$, the basic laws are

$$\begin{aligned} \lim_{x\to a}c &= c,\\ \lim_{x\to a}x &= a,\\ \lim_{x\to a}\bigl(f(x)\pm g(x)\bigr) &= L\pm M,\\ \lim_{x\to a}c f(x) &= cL,\\ \lim_{x\to a}f(x)g(x) &= LM,\\ \lim_{x\to a}\frac{f(x)}{g(x)} &= \frac{L}{M}\quad(M\ne0),\\ \lim_{x\to a}f(x)^n &= L^n. \end{aligned}$$

For $n$th roots,

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

provided the expression is defined near $a$; for an even root, this includes the usual nonnegative-domain restriction.

These laws combine limits that already exist. They do not prove the existence of a component limit, and indeterminate forms such as $0/0$ are signals to transform or analyze the expression rather than substitute mechanically.

Consequences

• Polynomials are continuous everywhere, so their finite limits are found by direct substitution.
• Rational functions are continuous wherever their denominators are nonzero.
• If $g(x)\le f(x)\le h(x)$ near $a$ and both outer functions approach the same value $L$, the squeeze theorem gives $\lim_{x\to a}f(x)=L$.