Chapter 2. Limits and rates of change
Section 2.3 Calculating limits using the limit laws
Limit laws Suppose that c is a constant and the limits lim f (x) and lim g(x) exist. Then
x→a
x→a
1. lim [f (x) + g(x)] = lim f (x) + lim g(x)
x→a
x→a
x→a
2. lim [f (x) − g(x)] = lim f (x) − lim g(x)
x→a
x→a
x→a
3. lim cf (x) = c lim f (x)
x→a
x→a
4. lim f (x)g(x) = lim f (x) · lim g(x)
x→a
x→a
x→a
lim f (x)
f (x)
= x→a
if lim g(x) ̸= 0
lim g(x) x→a
x→a g(x)
x→a
[
]n
6. lim [f (x)]n = lim f (x) where n is a positive integer
5. lim
x→a
x→a
7. lim c = c
x→a
8. lim x = a
x→a
9. lim xn = an where n is a positive integer
x→a
√
√
10. lim n x = n a where n is a positive integer
x→a
√
√
11. lim n f (x) = n lim f (x) where n is a positive integer
x→a
x→a
Example 1. Given that lim f (x) = 2, lim g(x) = −1, and lim h(x) = 10. Find the limits
x→a
x→a
x→a
that exist.
1. lim [2f (x) − g(x) − h(x)]
x→a
g(x)
x→a h(x) − 2f (x)
2. lim
Example 2. Evaluate the given limit and justify each step.
1. lim (2x2 + 4x − 1)
x→4
1
3(8y 2 − 1)
2
4
y→3 2y (y − 1)
2. lim
3. lim
x→3
√
4
x2 + 2x + 1
If f is a polynomial or a rational function and a is in the domain of f , then lim f (x) = f (a)
x→a
Example 3. Evaluate each limit, if it exist.
2
1. lim x x−+x 1− 2
x→−1
2
2. lim x x−+x 1− 3
x→−1
3
4. lim t2 − t
t→1 t − 1
5. lim 9 −√t
t→9 3 −
t
2
6. lim √
x→0
x
1 + 3x − 1
⟨
8. lim ⃗r(t), ⃗r(t) =
t→2
4 −√t , t2 − 4
2− t t−2
⟩
9. lim |x + 3|
x→−3
|x − 2|
10. lim x − 2
x→2
3
Example
Let
{ 4.
x2 − 2x + 2, if x < 1
f (x) =
3 − x,
if x ≥ 1
Find lim f (x).
x→1
Theorem If f (x) ≤ g(x) for all x in an open interval that contains a (except possibly at a)
and the limits of f an 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) for all x in an open interval that contains a
(except possibly at a) and lim f (x) = lim h(x) = L, then
x→a
x→a
lim g(x) = L
x→a
Example 5. Use the Squeeze Theorem to show that lim x2 cos(20πx) = 0.
x→0
4