Consider a certificate of deposit that pays 10% (annual percentage rate) on an initial deposit of $1000. The balance after 10 years is $A = 1000 ( 1 + 0 \cdot 1 x ) ^ { 10 / x }$ . Estimate $\lim _ { x \rightarrow 0 ^ { + } } A$ , where $x$ is the length of the compounding period (in years). Round your answer to the nearest hundredth.
A)2718.28
B)367.88
C)1000.00
D)10,000.00
E)1100.00

Question

Quizplus · Accepted Answer

The limit as $x \rightarrow 0^+$ of $A = 1000(1 + 0.1x)^{10/x}$ represents the scenario of continuous compounding interest. This is described by the formula $A = Pe^{rt}$, where $P = 1000$, $r = 0.1$, and $t = 10$. Substituting these values gives $A = 1000e^{(0.1)(10)} = 1000e^1 = 1000e$. The value of $e$ (Euler's number) is approximately 2.71828, so $A ≈ 1000 \times 2.71828 = 2718.28$.

Consider a Certificate of Deposit That Pays 10% (Annual Percentage A=1000(1+0⋅1x)10/xA = 1000 ( 1 + 0 \cdot 1 x ) ^ { 10 / x }A=1000(1+0⋅1x)10/x

Consider a Certificate of Deposit That Pays 10% (Annual Percentage $A = 1000 ( 1 + 0 \cdot 1 x ) ^ { 10 / x }$