1549901369-Elements_of_Real_Analysis__Denlinger_

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