Step-by-step explanation:
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)).
