174 CHAPTER 8. POWER SERIES
and f ′ and g′ exist on (a,b) with g′ (x) ̸= 0 on (a,b). Suppose also that
limx→a+
f ′ (x)g′ (x)
= L. (8.29)
Then
limx→a+
f (x)g(x)
= L. (8.30)
Theorems 8.8.1 8.8.8 and Corollaries 8.8.2 and 8.8.9 will be referred to as L’Hôpital’srule from now on. Theorem 8.8.1 and Corollary 8.8.2 involve the notion of indeterminateforms of the form 0
0 . Please do not think any meaning is being assigned to the nonsenseexpression 0
0 . It is just a symbol to help remember the sort of thing described by Theorem8.8.1 and Corollary 8.8.2. Theorem 8.8.8 and Corollary 8.8.9 deal with indeterminate formswhich are of the form ±∞
∞. Again, this is just a symbol which is helpful in remembering the
sort of thing being considered. There are other indeterminate forms which can be reducedto these forms just discussed. Don’t ever try to assign meaning to such symbols.
Example 8.8.10 Find limy→∞
(1+ x
y
)y.
It is good to first see why this is called an indeterminate form. One might think that asy→ ∞, it follows x/y→ 0 and so 1+ x
y → 1. Now 1 raised to anything is 1 and so it wouldseem this limit should equal 1. On the other hand, if x > 0, 1+ x
y > 1 and a number raisedto higher and higher powers should approach ∞. It really isn’t clear what this limit shouldbe. It is an indeterminate form which can be described as 1∞. By definition,(
1+xy
)y
= exp(
y ln(
1+xy
)).
Now using L’Hôpital’s rule,
limy→∞
y ln(
1+xy
)= lim
y→∞
ln(
1+ xy
)1/y
= limy→∞
11+(x/y)
(−x/y2
)(−1/y2)
= limy→∞
x1+(x/y)
= x
Therefore, limy→∞ y ln(
1+ xy
)= x. Since exp is continuous, it follows
limy→∞
(1+
xy
)y
= limy→∞
exp(
y ln(
1+xy
))= ex.
8.8.1 Interest Compounded ContinuouslySuppose you put money in the bank and it accrues interest at the rate of r per paymentperiod. These terms need a little explanation. If the payment period is one month, andyou started with $100 then the amount at the end of one month would equal 100(1+ r) =100+100r. In this the second term is the interest and the first is called the principal. Nowyou have 100(1+ r) in the bank. This becomes the new principal. How much will youhave at the end of the second month? By analogy to what was just done it would equal
100(1+ r)+100(1+ r)r = 100(1+ r)2 .