Download 2.2 Algebraic Computation of Limits Basic Properties and Rules for

2.2
Algebraic Computation of Limits
Basic Properties and Rules for Limits
For any real number c, suppose that f and g both have limits at x = c.
Constant Rule
lim k = k
Limit of x Rule
lim x = c
Multiple Rule
lim kf(x) = k lim f(x)
Sum Rule
lim [f(x) + g(x)] = lim f(x) + lim g(x)
Difference Rule
lim [f(x) - g(x)] = lim f(x) - lim g(x)
Product Rule
lim [f(x)g(x)] = [ lim f(x)][ lim g(x)]
Quotient Rule
lim
xc
xc
xc
xc
xc
xc
xc
xc
xc
xc
xc
xc
xc
lim f(x)
f(x)
= xc
lim g(x)
g(x)
xc
Power Rule


lim [f(x)]n = lim f(x)
xc
Example: Evaluate the limit
lim (x2 + 2x + 5)
x3
xc
xc
n
Limit of a Ploynomial Function
If P is a polynomial function, then
lim P(x) = P(c)
xc
Example: Evaluate the limit:
lim (x3 + 3x)
Example: Evaluate the limit:
lim
x 5
x 1
x2  5x - 2
Limit of a Rational Function
If Q is a rational function denoted by Q(x) =
lim Q(x) =
xc
P(x)
, then
D(x)
P(c)
D(c)
provided that lim D(x) ≠ 0.
xc
Example: Evaluate the limit:
lim
x 3
x2  3x - 10
3x2  5x  7
Limits of Trigonometric Functions
If c is any number in the domain of the given function, then
lim sin x = sin c
lim cos x = cos c
lim tan x = tan c
lim cot x = cot c
lim sec x = sec c
lim csc x = csc c
xc
xc
xc
xc
xc
xc
Using Algebra To Find Limits
Example: Evaluate the limit:
Example: Evaluate the limit:
lim
x2
lim
x 2
x2 - 4 x - 4
x2 - x - 2
x  2 -2
x -2
Special Limits Involving Sine and Cosine
lim
h0
sin h
=1
h
and
Example: Evaluate the limit:
lim
Example: Evaluate the limit:
lim
lim
h0
cos h - 1
=0
h
sin 4x
9x
x0
x 0
cot 3x
cot x
Squeeze Rule (or Sandwich Rule)
g(x) ≤ f(x) ≤ h(x) on an open interval containing c, and if
lim g(x) = lim h(x) = L
xc
xc
then
lim f(x) = L
xc