Question

(a) Use tables to give a numerical argument that $$\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{x}=e$$ Support your argument graphically. (b) For several different values of \(r\) give numerical and graphical evidence that $$\lim _{x \rightarrow \infty}\left(1+\frac{r}{x}\right)^{x}=e^{r}$$ (c) Writing to Learn Explain why compounding interest over smaller and smaller periods of time leads to the concept of interest compounded continuously.

Short answer

The numerical and graphical analysis show that as \(x\) approaches infinity, \((1 + 1/x)^{x}\) approaches \(e\), and for different \(r\)'s, \((1 + r/x)^{x}\) approaches \(e^{r}\). This concept is used in finance to compound interest continuously.

Step-by-step solution

Part (a) - Numerical Calculation

Create a table of values for \(x\) and \((1 + 1/x)^{x}\) for various large values of \(x\). As \(x\) increases, it could be observed that \((1 + 1/x)^{x}\) approximates the value of \(e\) which is roughly 2.71828.

Part (a) - Graphical Argument

On a graph, plot \(x\) against \((1 + 1/x)^{x}\). As \(x\) approaches infinity, the value of the function approaches the horizontal line \(y = e\), providing a graphical support for the limit.

Part (b) - Numerical and Graphical Evidence

Similar to part (a), create tables of values for various large values of \(x\) and for different \(r\)'s, and plot the functions \((1 + r/x)^{x}\) for each \(r\). It will be observed both numerically and graphically that \((1 + r/x)^{x}\) approximates \(e^{r}\) as \(x\) approaches infinity.

Part (c) - Understanding Continuous Compounding

In the context of financial mathematics, the expression \((1 + r/n)^{nt}\) usually represents the compound interest attained after compounding \(n\) times per year over \(t\) years. When the compounding is increased, that is, as \(n\) becomes larger and larger (reaches infinity), this expression approaches \(e^{rt}\), which is the concept of continuous compounding. This is the limit that banks use to compound interest continually, hence the concept of interest compounded continuously.