Q: Which of the following is the correct limit definition of $e$?

$e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$

Question 1

Which of the following best describes the mathematical constant $e$?

Accepted Answer

$e$ is an irrational number that is the base of natural logarithms.

Question 2

Which formula gives the amount $A$ in an account after $t$ years if $P$ dollars are invested at an annual rate $r$ compounded continuously?

Accepted Answer

$A = Pe^{rt}$

Question 3

Suppose $1,000 is invested at 6\%$ interest compounded continuously. What is the balance after 5 years? (Round to the nearest cent.)

Accepted Answer

$A = 1000e^{0.3} \approx 1349.86$

Question 4

A principal of $5,000 is invested at 6\%$ compounded continuously. How long will it take for the investment to grow to $8,000$?

Accepted Answer

$t = \frac{\ln(8,000/5,000)}{0.06} \approx 7.87$ years

Question 5

If an investment is compounded continuously at a rate $r$, what is the formula for the doubling time?

Accepted Answer

$t = \frac{\ln 2}{r}$

Question 6

A bank offers a 5-year certificate of deposit (CD) that earns 5.75\% compounded continuously. If $1,000 is invested, what is the amount in the account after 5 years?

Accepted Answer

$A = 1,000e^{0.2875} \approx 1,333.10$

Question 7

Which of the following best explains why continuous compounding yields more interest than daily compounding?

Accepted Answer

Continuous compounding applies interest at every instant, maximizing the effect of exponential growth.

Question 8

If $P = 2,000$, $r = 4\%$, and $t = 10$ years, what is the amount $A$ with continuous compounding?

Accepted Answer

$A = 2,000e^{0.4} \approx 2,983.64$

Question 9

A business invests $10,000$ at 7\% compounded continuously. How long will it take for the investment to triple?

Accepted Answer

$t = \frac{\ln 3}{0.07} \approx 15.7$ years

Question 10

Which of the following is the correct limit definition of $e$?

Accepted Answer

$e = \lim_{n 	o \infty} \left(1 + \frac{1}{n}ight)^n$

Question 11

A company wants to know how long it will take for an investment to quadruple at 8\% compounded continuously. What is the correct formula and answer?

Accepted Answer

$t = \frac{\ln 4}{0.08} \approx 17.33$ years

Question 12

If $1,000$ is invested at 5\% compounded continuously, how much interest is earned after 20 years?

Accepted Answer

$A = 1,000e^{1} \approx 2,718.28$, so interest = $1,718.28$

Question 13

Which scenario best illustrates the advantage of continuous compounding over annual compounding?

Accepted Answer

A retirement fund grows faster with continuous compounding because interest is added at every instant.

Question 14

If $A = Pe^{rt}$, which variable represents the annual interest rate?

Accepted Answer

$r$

Question 15

A principal of $2,500$ is invested at 3.5\% compounded continuously. What is the balance after 12 years?

Accepted Answer

$A = 2,500e^{0.42} \approx 3,726.25$

The Constant e and Continuous Compound Interest-Chapter 3

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