Question

Evaluate the following limits, if necessary use l’Hôpital Rule:

If an initial amount A₀ of money is invested at an interest rate r compounded n times a year, the value of the investment after t years is A = `"A"_0 (1 + "r"/"n")^"nt"`. If the interest is compounded continuously, (that is as n → ∞), show that the amount after t years is A = A₀e^rt 

Answer 1

A = `"A"_0 (1 + "r"/"n")^"nt"`  ......[A₀ is a constant]

Let y = `lim_(x -> oo) (1 + "r"/"n")^"nt"`

Taking log on both sides,

log y = `lim_(x -> oo) "nt" log(1 + "r"/"n")`  ......`[oo xx 0  "Indeterminate form"]`

= `lim_("n" -> oo) (log(1 + "r"/"n"))/(1/"nt")` ........`[0/0  "indeterminate form"]`

Applying L’ Hôpital’s Rule

= `lim_("n" -> oo) ((1/(1 + "r"/"n"))(- "r"/"n"^2))/(- 1/("n"^2"t"))`

log y = `lim_("n" -> oo) "rt"/(1 + "r"/"n")` = rt

Exponentating, we get

y = `"e"^pi`

We have A = `"A"_0 (1 + "r"/"n")^"nt"`

∴ A = `"A"_0  "e"^pi`

Hence Proved.