1. IF c is constant and lim f(x) and lim g(x) exist then…
Calc Limit Laws
IF c is constant and lim f(x) and lim g(x) exist then…
x->a
1. lim [f(x) + g(x)] = lim f(x) + lim g(x)
x->a
“The limit of a sum (or diff) is the sum (or diff) of the limits”
“The limit of a constant times a function is the constant times the limit of the function.”
2. lim [cf(x)] = c lim f(x)
x->a
x->a
3. lim [f(x)g(x)] = lim f(x) • lim g(x)
“The limit of a product is the product of the limits”
(= 0)
lim f(x)
lim g(x)
x->a
f(x)
g(x)
4. lim =
“The limit of a quotient is the quotient of the limits”

2. ex: Use the limit laws and the graphs of f and g below to evaluate the following limits if they exist…
a) lim [f(x) + 5g(x)]
x->-2 f g
b) lim [f(x)g(x)]
x->1
c) lim
x->2
f(x)
g(x)

3. Additional Laws lim [f(x)] = lim f(x) n 5. Power Law:
x->a n
5. Power Law:
6. Special Limits:
lim x = a
x->a n
lim x = a
x->a n
lim c = c
x->a
lim x = a
x->a
7. Root Law:
lim f(x) = lim f(x)
x->a n
(where n is a positive integer)

4. “If f is a polynomial or rational function and a is in the domain of
f then”:…
lim f(x) = f(a)
x->a
Direct Substitution
ex: lim 2x2 - 3x + 4 = [2(5)2 - 3(5) + 4] = 39
x->5
“Continuous at a”
(x + 1)(x - 1)
x - 1
lim
x->1
lim
x->1
x2 - 1
x - 1 =
lim (x + 1) = 2
x->1 =

5. (3 + h)2 = (3 + h)(3 + h) = [9 + 6h + h2]
ex: Evaluate lim
h-->0
(3 + h)2 - 9 h
- 9
(3 + h)2 = (3 + h)(3 + h) = [9 + 6h + h2]
6h + h2 h =
lim
h-->0 =
h (6 + h) h
lim
h-->0 =
lim
h-->0
(6 + h)
= 6
t2 + 9
ex: Find lim
t-->0
- 3 t2
t2 + 9
+ 3 =
(t2 + 9) - 9 t2
t2 + 9
+ 3
t-->0
lim = t2
t-->0
lim
t2 + 9
+ 3 =
t-->0
lim 1
t2 + 9
+ 3 1 =
t-->0
lim
9 + 3 6

6. If f(x) < g(x) when x is near a (except possibly at a), and the limits of f and g both exist as x approaches a, then:
lim f(x) < lim g(x)
x-->a
x-->a
The Squeeze Theorem
If f(x) < g(x) < h(x) when x is near a (except possibly at a) and:
lim f(x) = lim h(x) = L
x-->a
x-->a
then…
lim g(x) = L
x-->a