4.3 THE LIMIT CONCEPT (OPTIONAL) 137(b)
Figure 4.6 We can prow 0 # lim,,, f (x), btrt the
specific due of E we use in the proof my not exceed 1.
(a) Any E > 1 is too large to be useful in proving
0 # lim f(x) as x tends to 0; (b) E = 4 does the job.
axis. This, of course, means that the proposed limit does not exist, as
our intuition had indicated from the start. In approaching this prob-
lem, you should separate the problem into three cases L 5 - 1,- 1 c L c 1, and L 2 1 (see Exercise 3(b)). 0
2x + a,
EXAMPLE 3 Let f (x) = '1, where o is a hred positive real
x-a, x<O
number. Argue that 0 # lim,,, f (x).