Basic Limit Laws

Listed here are a couple of basic limits and the standard limit laws which, when used in conjunction, can find most limits. They are listed for standard, two-sided limits, but they work for all forms of limits. However, note that if a limit is infinite, then the limit does not exist.

Basic Limits

If c is a constant, then the limit as x goes to a of c is c.

the limit as x goes to a of x is a.

Limit Laws

Addition Law

If the limits the limit as x goes to a of f(x) and the limit as x goes to a of g(x) both exist, then

the limit as x goes to a of f(x)+g(x) equals the limit as x goes to a of f(x) + the limit as x goes to a of g(x).

Subtraction Law

If the limits the limit as x goes to a of f(x) and the limit as x goes to a of g(x) both exist, then

the limit as x goes to a of f(x)-g(x) equals the limit as x goes to a of f(x) - the limit as x goes to a of g(x).

Constant Law

If c is a constant, and the limit the limit as x goes to a of f(x) exists, then

the limit as x goes to a of c*f(x) = c* the limit as x goes to a of f(x).

Multiplication Law

If the limits

the limit as x goes to a of f(x)

and

the limit as x goes to a of g(x)

both exist, then

the limit as x goes to a of f(x)*g(x) = the limit as x goes to a of f(x) * the limit as x goes to a of g(x)

.

Division Law

If the limits the limit as x goes to a of f(x) and the limit as x goes to a of g(x) both exist, and the limit as x goes to a of g(x) is not zero, then

the limit as x goes to a of (f(x)/g(x)) = (the limit as x goes to a of f(x))/(the limit as x goes to a of g(x)).

Power Law

If n is an integer, and the limit the limit as x goes to a of f(x) exists, then

the limit as x goes to a of (f(x))^n = (the limit as x goes to a of f(x))^n.

Root Law

If n is an integer, the limit the limit as x goes to a of f(x) exists, and that limit is positive if n is even, then

the limit as x goes to a of the nth root of f(x) = the nth root of the limit as x goes to a of f(x).

Squeeze Law

If f(x) <= g(x) <= h(x) for all x in an open interval that contains a, except possibly at a itself, and the limit as x goes to a of f(x) = the limit as x goes to a of h(x) = L, then

the limit as x goes to a of g(x) = L.

Composition Law

If f is continuous at b and the limit as x goes to a of g(x) = b, then

the limit as x goes to a of f(g(x)) = f(b) = f(the limit as x goes to a of g(x)).

Copyright © 1996 Department of Mathematics, Oregon State University

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