Bridge to Abstract Mathematics: Mathematical Proof and Structures

4.3 THE LIMIT CONCEPT (OPTIONAL) 131

[namely, f(2) = 81. The problem is that the two values, 7 and 8, are dif-

ferent; this f has a-removable discontinuity at x = 2. An important missing

link in this discussion, and a question you should be thinking about, is

"Why, according to the epsilon-delta definition of limit, does the preceding

limit equal 7?'A careful analysis of the epsilon-delta definition, soon to

come, provides the answer.

(b) A typical example for which lim,,, f(x) exists, while f(a) does

not exist, is a function defined by a rule such as

which collapses to 0/0, and is left undefined, at x = 6. For all values of x
except 6, f(x) equals x + 6. In fact, limXd6 (x2 - 36)/(x - 6) = limx+6 x + 6
= 12. [Again, when graphing this function, note that the limit is the y
component of the "missing point" (6, 12)]. Thus the limit exists even though
f(6) is undefined. Again, we have a removable discontinuity, this time at
x = 6.
This category of limit problem is particularly important because, every
time we use the definition f'(a) = lirn,,, (f(x) - f(a))/(x - a) of the deriva-
tive, we must deal with a limit of this type. Review how to compute d/dx(x3)
and d/dx(x4 + x2) from the definition of derivative.

Type 111. Under this category, we consider situations in which lirn,,, f(x)

does not exist. Once again, f may or may not be defined at a. This category

may be divided into three subcases:

One-sided (i.e., left- and right-hand) limits exist, but are different. For

example, if

we have lim,,,- f(x) = -1 # 1 = lim,,,+ f(x). Thus lirn,,, f(x)

does not exist. (Note: an important theorem, which you should recall

from calculus, states that lirn,,, f(x) exists if and only if lim,,,- f(x)

and lirn,,, + f (x) both exist and are equal.]

Infinite limits at x = a (i.e., the line x = a is a vertical asymptote to the

curve). This occurs especially in a rational function in which the de-

nominator tends to zero as x -+ a while the numerator approaches

some nonzero quantity. This fact will be proved in Article 6.2, when

we consider indirect proofs (see Exercise 16, Article 6.2.).

The one-sided limits are not infinite. but thev do not exist as finite real

numbers either. This is the strangest case, since it must involve rather

wild fluctuations in the function. An example of this is f(x) = sin (llx),

x # 0. In any interval (0, p) this function "bounces up and down" be-

tween y = - 1 and y = 1 infinitely many times, no matter how small

we choose the positive real number p.