Chapter 2 | Limits
141
2.5 Use the graph of h ( x ) in Figure 2.16 to evaluate lim x →2
h ( x ), if possible.
Figure 2.16
Looking at a table of functional values or looking at the graph of a function provides us with useful insight into the value of the limit of a function at a given point. However, these techniques rely too much on guesswork. We eventually need to develop alternative methods of evaluating limits. These new methods are more algebraic in nature and we explore them in the next section; however, at this point we introduce two special limits that are foundational to the techniques to come.
Theorem 2.1: Two Important Limits Let a be a real number and c be a constant.
lim x lim x
x = a
(2.4) (2.5)
i.
→ a
c = c
ii.
→ a
We can make the following observations about these two limits. i. For the first limit, observe that as x approaches a , so does f ( x ), because f ( x ) = x . Consequently, lim x → a x = a . ii. For the second limit, consider Table 2.4 . x f ( x ) = c x f ( x ) = c
a +0.1
a −0.1
c
c
a +0.01
a −0.01
c
c
a +0.001
a −0.001
c
c
a +0.0001
a −0.0001
c
c
Table 2.4 Table of Functional Values for lim x → a
c = c
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