Question

The Definition of e Illustrate the definition of the number e by graphing the curve$y={\left(1+\frac{1}{x}\right)}^x$and the line$y=e$on the same screen, using the viewing rectangle $[0,40]$by $[0,4]$.

Short answer

On the graph $y={\left(1+\frac{1}{x}\right)}^x$, as x approaches $\infty$, then y approaches to e.

So, the definition of e is: $e=\underset{x\to \infty }{\lim }{\left(1+\frac{1}{x}\right)}^x$.

Step-by-step solution

Step 1. Given.

The given curve is: $y={\left(1+\frac{1}{x}\right)}^x$. The given line is $y=e$.

Step 2. To determine.

We have to illustrate the definition of the number e by graphing the curve $y={\left(1+\frac{1}{x}\right)}^x$ and the line$y=e$ on the same screen, using the viewing rectangle $[0,40]\mathrm{by}[0,4]$.

Step 3. Calculation.

We’ll use a graphing calculator to graph $y={\left(1+\frac{1}{x}\right)}^x$and $y=e$on the same screen, using the viewing rectangle $[0,40]\mathrm{by}[0,4]$.

From the graph, the line$y=e$ is a horizontal asymptote to the curve $y={\left(1+\frac{1}{x}\right)}^x$.

It means, the limit$x\to \infty$ of the function$y={\left(1+\frac{1}{x}\right)}^x$ is e.

Or, on the graph $y={\left(1+\frac{1}{x}\right)}^x$, as x approaches $\infty$, then y approaches to e.

So, the definition of e is: $e=\underset{x\to \infty }{\lim }{\left(1+\frac{1}{x}\right)}^x$.