4.1 Definition of Limit for Functions 185'Vo> 0, 3x E V(f) 3 0 < Ix - xol < o but lf(x) - LI ~ c. Keep this c fixed
throughout the remainder of the proof. Taking 8 = 1, ~, ~, · · · ~, · · · we have
:lx1 E V(f) 3 0 < lx1 - xol < 1 but lf(xi) - LI~ c;
:lx2 E V(f) 3 0 < lx2 - xol <~but lf(x2) - LI~ c;3 xn E V(f) 3 0 < lxn - xol <~but lf(xn) - LI~ c;
In this way we generate a sequence { Xn} in V(f) such that 't/n E N, 0 <
lxn-xol <~but lf(xn)-LI ~ c. Thus, {xn} is a sequence in V(f)-{xo} and,
by the squeeze principle, Xn ___, xo. On the other hand, the sequence {f (xn)}
cannot converge to L since 'Vn E N, lf(xn) - LI ~ c. This contradicts our
hypothesis. Therefore, lim f(x) = L. •
x--+xo
USING THE SEQUENTIAL CRITERION TO
DISPROVE lim f(x) = L
ro-+rooCorollary 4.1.10 If 3 sequence { Xn} in V(f) - { xo} such that Xn ___, xo, but
the sequence {f(xn)} does not converge to L, then f(x) does not have limit L
at xo.
Proof. This is the contrapositive of Theorem 4.1.9. •Corollary 4.1.11 If 3 sequences {xn} and {yn} in V(f) - {xo}, which both
converge to x 0 , but the sequences {f (xn)} and {f (yn)} do not both converge to
the same number, then lim f(x) does not exist.
X--+XQ
Proof. Immediate consequence of Theorem 4.1.9 and the uniqueness of
limits of functions. •
Example 4.1.12 Prove^4 that lim sin(~) does not exist.^5 [See Figure 4.2.]
x->O X
- For an alternate proof, see Exercise 11.
- We shall define sin x formally in Chapters 7 and 9. Until then, we assume the usual a lgebraic
properties of sin x.