4.2 Algebra of Limits of Functions 195
Finally, we apply the algebra of limits again to conclude that
n
lim p(x) = lim 2:: akxk
X->Xo X->Xo k=O
= f= ( lim akxk)
k=O X->Xo
n
= L akx~
k=O
= p(xo). •
Example 4.2.14 lim(3x^3 - 7x^2 + x + 11) = 3(8) - 7(4) + 2+11 = 9.
X->2
Note: Theorem 4.2.13 tells us that limits are of no essential significance in
the study of polynomials. The limit of a polynomial at xo is found by merely
"plugging in" x 0. Obviously, limits were introduced to study functions more
complicated than polynomials.
Definition 4.2.15 A rational function (of one variable) is any function of
the form
p(x)
r(x) = q(x),
where p(x) and q(x) a re polynomials.
Theorem 4.2.16 (Limits of Rational Functions) For any rational func-
tion r(x) = p((x)), and any x 0 E IR, lim r(x) = r(xo) provided that q(xo) =f-0.
q X X->Xo
Proof. Apply Theorem 4 .2.ll(f) and Theorem 4.2. 13. •
5x - 3 12
Example 4.2.17 ~~
2
x 2 +
1 19
In Chapter 2 we saw that in discussing conver gence of a sequence, "only
the tail matters." (See Theorem 2.2. 16 .) Similarly, in discussing the limit of
a function as x ___, x 0 , only what h appens in a deleted neighborhood of xo
matters. That principle is formalized in the next theorem.
Theorem 4.2.18 (Only What Happens in a Deleted Neighborhood of
Xo Matters) Suppose lim f(x) = L , and f(x) = g(x) for all x in some
x-.xo
deleted neighborhood of xo. Then lim g(x) = L.
X-+XQ