3.3 Unilateral limits and asymptotes 1333.3 Unilateral limits and asymptotes
the function f for which f(x) = sgn (x - a) and see the graph shown in
Figure 3.31, and in some other cases as well,
we can cheerfully assertthat AR (lambda sub
R) is a number such that f(x) is near AR when-
ever x is near a and x > a. We can feel sure i a x
that we know the meaning of the assertion, but
we must nevertheless know that the epsilon- Figure 3.31
delta version of the assertion is the following.
To each e > 0 there corresponds a S > 0 such that lf(x) - ARJ < e
whenever a < x < a + 3. This time the condition x 74- a does not enter
the assertion to bother our little sister and everything is very simple.
The abbreviated version of the assertion is
(3.32) lim f(x) = X.
x-+a+
The new thing in this symbol is the plus sign that follows the a. Perhaps
the best way to read this is "the right-hand limit as x approaches a of
f (x) is XR," but it is always awkward to write one thing and say another,
so the reading usually boils down to "the limit as x approaches a plus of
f(x) is AR." In case there is no number for which the assertion is valid,
we say that the right-hand limit does not exist. A similar succession of
ideas leads to the symbol(3.321) lira f(x) = AL,
x-+a-
which says that the left-hand limit as x approaches a of f(x) is AL.
If a function f and a number xo are such that the unilateral limits AR
and AL in(3.33) lim f(x) = AR, lira f(x) = AL
X- xo+exist and are different, then the function f is said to have a jump (or an
ordinary discontinuity) at the point xo. The magnitude of the jump is
JAR - AL!. If AR > AL, then f has an upward jump, and if AR < AL, then
f has a downward jump.
Another assertion that turns out to be both interesting and important
is the assertion that a function f and a number L may be such that f(x)
is near L whenever x is large. When making this assertion precise, we do
not use the letters a and 3 but, instead, use a and some other letter, say
N, that we can easily regard as a "large" number. The assertion means
that to each e > 0 there corresponds a number N such that(3.34) 1f(x) - Li < e (x > N).
By tossing in some surplus verbiage, we can put this in terms that may be