Page 73 - Calculus Demystified

CHAPTER 2
60

Fig. 2.3 Foundations of Calculus

EXAMPLE 2.3
Define

3if x = 7
h(x) =
1if x = 7

Calculate lim x→7 h(x).

SOLUTION
It would be incorrect to simply plug the value 7 into the function h and
thereby to conclude that the limit is 1. In fact when x is near to 7 but unequal
to 7, we see that h takes the value 3. This statement is true no matter how close
x is to 7. We conclude that lim x→7 h(x) = 3.

2
You Try It: Calculate lim x→4 [x − x − 12]/[x − 4].

2.1.1 ONE-SIDED LIMITS
There is also a concept of one-sided limit. We say that

lim f(x) =
x→c −
if the values of f become closer and closer to when x is near to c but on the left.
In other words, in studying lim x→c − f(x), we only consider values of x that are
less than c.
Likewise, we say that
lim f(x) =
x→c +
if the values of f become closer and closer to when x is near to c but on the right.
In other words, in studying lim x→c + f(x), we only consider values of x that are
greater than c.