128 CHAPTER 7. THE DERIVATIVE

|y− x| < δ ,y ∈ D( f ) , and yet | f (y)−L| ≥ ε . Now let δ 1 = 1. There exists x1 ̸= x withx1 ∈ B(x,δ 1)∩D( f ) and | f (x1)−L| ≥ ε. Let δ 2 ≡ min

( 12 ,

12 |x− x1|

). Now pick x2 ∈

B(x,δ 2) ,x2 ̸= x such that | f (x2)−L| ≥ ε. Let δ 3 ≡ min(

123 ,

12 |x− x1| , 1

2 |x− x2|)

andpick x3 ∈ B(x,δ 3) with | f (x3)−L| ≥ ε,x3 ̸= x. Continue this way to generate a sequenceof distinct points {xn} , none equal to x which converges to x. Then L = limn→∞ f (xn) be-cause of the condition on limits of the sequence so eventually |L− f (xn)|< ε, contrary tothe construction of the xn.

The value of a function at x is irrelevant to the value of the limit at x! This mustalways be kept in mind. In fact, it is not necessary for f to even be defined at the limit point.All interesting limits are this way. You do not evaluate interesting limits by computingf (x)! It may be the case that f (x) is right but this is merely a happy coincidence when itoccurs and as explained below in Theorem 7.1.8, this is sometimes equivalent to f beingcontinuous at x.

Theorem 7.1.5 In this theorem, x is always a limit point of D( f ). Suppose that bothlimy→x f (y) = L and limy→x g(y) = K where K and L are numbers, not ±∞. Then if a, bare numbers,

limy→x

(a f (y)+bg(y)) = aL+bK, (7.1)

limy→x

f g(y) = LK (7.2)

and if K ̸= 0,

limy→x

f (y)g(y)

=LK. (7.3)

Also, if h is a continuous function defined in some interval containing L, then

limy→x

h◦ f (y) = h(L) . (7.4)

Suppose f is real valued and limy→x f (y) = L. If f (y) ≤ a all y near x either to the rightor to the left of x, then L≤ a and if f (y)≥ a then L≥ a.

Proof: All of these claims follow from Proposition 7.1.4 and the theorems on limits ofsequences Theorem 4.4.8. For example, consider 7.4. Letting x be a limit point and xn→ x,then by assumption f (xn)→ L and so, by continuity of h, it follows that h( f (xn))→ h(L).If f (y) ≤ a for all y less than a and near to x, then if xn → x from the left, eventuallyf (xn)≤ a and so L≤ a also. The other case is similar.

A very useful theorem for finding limits is called the squeezing theorem.

Theorem 7.1.6 Suppose f ,g,h are real valued functions and that

limx→a

f (x) = L = limx→a

g(x)

and for all x near a, f (x)≤ h(x)≤ g(x). Then limx→a h(x) = L.

Proof: If L ≥ h(x) , then |h(x)−L| ≤ | f (x)−L| . If L < h(x) , then |h(x)−L| ≤|g(x)−L| . Therefore,

|h(x)−L| ≤ | f (x)−L|+ |g(x)−L| .