Chapter 3.1, Problem 30E

a.

To determine

Show that the value of f(x) approaches the value of g(x) by graphically.

Explanation of Solution

Given:

It is given in the question that the function are $f(x)=1+{(\frac{3}{x})}^x$ and $g(x)=e^3$ also x increases without bound.

Concept Used:

In this,use the graph calculator and use the knowledge to understand the graph carefully and by this show it.

Proof:

The graph of $f(x)=1+{(\frac{3}{x})}^x$ and $g(x)=e^3$ are given below in same window.

The red line show the graph of $f(x)=1+{(\frac{3}{x})}^x$ .

The blue line show the graph of $g(x)=e^3$ .

From the graph , it is clearly seen that the curve of g(x) overlaps or approaches the curve of f(x).

So by this it proves that the value of f(x) approaches the value of g(x) .

  

b.

To determine

Show that the value of f(x) approaches the value of g(x) by numerically.

Explanation of Solution

Given:

It is given in the question that the function are $f(x)=1+{(\frac{3}{x})}^x$ and $g(x)=e^3$ also x increases without bound.

Concept Used:

In this , use the concept of changing the function by suitable arrangement and also use the concept ${(1+\frac{1}{x})}^x\to easx\to \infty$ .

Proof: There are two functions $f(x)=1+{(\frac{3}{x})}^x$ and $g(x)=e^3$ .

As it is know that ${(1+\frac{1}{x})}^x\to easx\to \infty$

Now, the f(x) can be rewrite as:

  $\begin{array}{l}$ f(x)={(1+\frac{3}{x})}^x={(1+\frac{1}{\frac{x}{3}})}^x$$ ={[{(1+\frac{1}{\frac{x}{3}})}^{\frac{x}{3}}]}^3$Let\frac{x}{3}=y\\ $ ={[{(1+\frac{1}{y})}^y]}^3\to e^3(x\leftrightarrow y)$\end{array}$