To find the value that $1 will approach with continuously compounding annual interest of 100% after one year, we use the formula for continuous compounding:

$A = P \times e^{rt}$

where:

• $A$ is the amount after time $t$,
• $P$ is the principal (initial amount),
• $r$ is the annual interest rate,
• $t$ is the time in years,
• $e$ is Euler's number, an irrational number approximately equal to 2.71828.

Given the problem:

• $P = 1$ (starting with $1),
• $r = 1$ (100% annual interest rate),
• $t = 1$ year.

Plugging in the values:

$A = 1 \times e^{1 \times 1} = e$

After one year, with continuous compounding at a 100% interest rate, the amount approaches $e$ (Euler's number).

So, the correct answer is:

(b) $e$

Would you like further details or have any questions?

Follow-Up Questions:

1. What would be the amount if the interest rate was 50% instead of 100%?
2. How does continuous compounding differ from annual compounding?
3. What is the significance of the number $e$ in mathematics and finance?
4. How would the amount change if the time period was 2 years instead of 1 year?
5. What if the interest rate was compounded quarterly instead of continuously?

Tip: Continuous compounding is an idealized way of calculating interest that assumes interest is being added constantly, leading to the mathematical constant $e$, which is central to many areas of mathematics and natural growth processes.