150 CHAPTER 5. THE DERIVATIVE
Theorems 5.17.1 5.17.8 and Corollaries 5.17.2 and 5.17.9 will each be referred to asL’Hò‚pital’s rule from now on. Theorem 5.17.1 and Corollary 5.17.2 involve the notion ofindeterminate forms of the form 0
0 . Please do not think any meaning is being assigned tothe nonsense expression 0
0 . It is just a symbol to help remember the sort of thing describedby Theorem 5.17.1 and Corollary 5.17.2. Theorem 5.17.8 and Corollary 5.17.9 deal withindeterminate forms which are of the form ±∞
∞. Again, this is just a symbol which is
helpful in remembering the sort of thing being considered. There are other indeterminateforms which can be reduced to these forms just discussed. Don’t ever try to assign meaningto such symbols.
Example 5.17.10 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’Hò‚pital’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.
5.18 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 equalthe expression 100(1+ r)+100(1+ r)r = 100(1+ r)2 . In general, the amount you wouldhave at the end of n months is 100(1+ r)n .
When a bank says they offer 6% compounded monthly, this means r, the rate perpayment period equals .06/12. Consider the problem of a rate of r per year and com-pounding the interest n times a year and letting n increase without bound. This is whatis meant by compounding continuously. The interest rate per payment period is then r/nand the number of payment periods after time t years is approximately tn. From the abovethe amount in the account after t years is P
(1+ r
n
)nt. Recall from Example 5.17.10 that
limy→∞
(1+ x
y
)y= ex. Then P
(1+ r
n
)nt can be written as P[(
1+ rn
)n]t and so, taking the
limit as n → ∞, you get Pert = A. This shows how to compound interest continuously.