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2. a. Many textbooks determine the formula for continuously compounded interest through an argument which avoids the use of I'Hopital's rule. Consider the compound amount A that satisfies the equation A = lim,, _ P(1+ :)". Let h = r. Then P ( 1 + = ) "- P ( 1 + h ) ( 1 / 1 )rt and we can focus on finding the limp o(1 + h)1/2. Show that (1 + h) 1/h _(1/h) In(1+h) and take the limit of both sides as / - 0. Hint: You can use the definition of the derivative in the exponent on the right-hand side. b. Consider an investment of amount P now and receive a sequence of positive payoffs { Al, A2, . .., An} at regular intervals. Use the Mean Value Theorem to show the rate of return defined by the root of the following function is unique: f (r ) = -P+ _A.(1 +r ) - Note: We can see that f(r) is continuous on the open interval (-1, co). In the limit as r approaches -1 from the right, the function values approach positive infinity. On the other hand as r approaches positive infinity, the function values approach -P