1.7. LIMITS, CONTINUITY, AND DIFFERENTIABILITY
67
from the left and write
lim
x→a −
f (x) = L 1
provided that we can make the value of f (x) as close to L 1 as we like by taking x
sufficiently close to a while always having x < a. In this case, we call L 1 the left-hand limit
of f as x approaches a. Similarly, we say L 2 is the right-hand limit of f as x approaches a
and write
lim
x→a +
f (x) = L 2
provided that we can make the value of f (x) as close to L 2 as we like by taking x sufficiently
close to a while always having x > a. In the graph of the function f in Figure 1.36, we see
that
lim
x→1 −
f (x) = 2 and lim
x→1 +
f (x) = 3
and precisely because the left and right limits are not equal, the overall limit of f as x → 1
fails to exist.
1
2
3
f
1
1
2
3
g
Figure 1.36: Functions f and g that each fail to have a limit at a = 1.
For the function g pictured at right in Figure 1.36, the function fails to have a limit at
a = 1 for a different reason. While the function does not have a jump in its graph at a = 1,
it is still not the case that g approaches a single value as x approaches 1. In particular, due
to the infinitely oscillating behavior of g to the right of a = 1, we say that the right-hand
limit of g as x → 1 + does not exist, and thus lim
x→1
g(x) does not exist.
To summarize, anytime either a left- or right-hand limit fails to exist or the left- and
67
from the left and write
lim
x→a −
f (x) = L 1
provided that we can make the value of f (x) as close to L 1 as we like by taking x
sufficiently close to a while always having x < a. In this case, we call L 1 the left-hand limit
of f as x approaches a. Similarly, we say L 2 is the right-hand limit of f as x approaches a
and write
lim
x→a +
f (x) = L 2
provided that we can make the value of f (x) as close to L 2 as we like by taking x sufficiently
close to a while always having x > a. In the graph of the function f in Figure 1.36, we see
that
lim
x→1 −
f (x) = 2 and lim
x→1 +
f (x) = 3
and precisely because the left and right limits are not equal, the overall limit of f as x → 1
fails to exist.
1
2
3
f
1
1
2
3
g
Figure 1.36: Functions f and g that each fail to have a limit at a = 1.
For the function g pictured at right in Figure 1.36, the function fails to have a limit at
a = 1 for a different reason. While the function does not have a jump in its graph at a = 1,
it is still not the case that g approaches a single value as x approaches 1. In particular, due
to the infinitely oscillating behavior of g to the right of a = 1, we say that the right-hand
limit of g as x → 1 + does not exist, and thus lim
x→1
g(x) does not exist.
To summarize, anytime either a left- or right-hand limit fails to exist or the left- and