Chapter 3.1, Problem 29E

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 $f(x)=1+{(\frac{0.5}{x})}^x$ and $g(x)=e^{0.5}$ 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 $f(x)=1+{(\frac{0.5}{x})}^x$ and $g(x)=e^{0.5}$ are given below in same window.

The red line show the graph of $f(x)=1+{(\frac{0.5}{x})}^x$ .

The blue line show the graph of $g(x)=e^{0.5}$ .

From the graph , it is clearly seen that the curve of g(x) overlaps 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 $f(x)=1+{(\frac{0.5}{x})}^x$ and $g(x)=e^{0.5}$ 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 $f(x)=1+{(\frac{0.5}{x})}^x$ and $g(x)=e^{0.5}$ .

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{0.5}{x})}^x={(1+\frac{1}{2x})}^x$$ ={[{(1+\frac{1}{2x})}^{2x}]}^{0.5}$Let2x=y\\ $ ={[{(1+\frac{1}{y})}^y]}^{0.5}\to e^{0.5}=\sqrt{e}(x\leftrightarrow y)$\end{array}$