2. One-sided limit
Given a function defined by 𝑦 = 𝑓 𝑥
lim
𝑥→𝑎+
𝑓 𝑥
Is the value y (real number) being approached by
f(x) as x gets closer and closer to a from the
right.
3. One-sided limit
Given a function defined by 𝑦 = 𝑓 𝑥
lim
𝑥→𝑎−
𝑓 𝑥
Is the value y (real number) “being approached”
by f(x) as x gets closer and closer to a from the
left.
6. One-sided limit
3
92
x
x
xf
x
3
92
x
x
xf x
3
92
x
x
xf
-3.1 -6.1 -2.9 -5.9
-3.01 -6.01 -2.99 -5.99
-3.001 -6.001 -2.999 -5.999
-3.0001 -6.0001 -2.9999 -5.9999
Table 2.1 Some numerical computations close to -3
from the left and right
7. One-sided limit
3
92
x
x
xf
Theorem 2.1 Existence of Limit
The limit of a function exists if and only if the one-
sided limits of the function are equal
lim
𝑥→𝑎+
𝑓 𝑥 = lim
𝑥→𝑎−
𝑓 𝑥
8. One-sided limit
Given a function defined by
𝑓 𝑥 =
𝑥2
− 9
𝑥 + 3
lim
𝑥→−3−
𝑓 𝑥 = lim
𝑥→−3+
𝑓 𝑥
lim
𝑥→−3
𝑓 𝑥 = −6