Chapter D, Problem 18E

To determine

To calculate:

Value of $N$ by using definition $2$ .

Answer to Problem 18E

Value of $N$ by using definition $2$ is $N=3$ and $N=19$ .

Explanation of Solution

Given information:

  $\underset{x\to \infty }{\lim }\frac{\sqrt{4x^2+1}}{x+1}=2$

  $\epsilon =0.5,\epsilon =0.1$

Calculation:

Let, $f$ be a function defined on some interval $\left(a,\infty \right)$ then,

  $\underset{x\to \infty }{\lim }f\left(x\right)=L$

Means that for every number $\epsilon >0$ there is corresponding number $N$ such that

If $x>N$ then $\left|f\left(x\right)-L\right|<\epsilon$

Given function is:

  $f\left(x\right)=\frac{\sqrt{4x^2+1}}{x+1}$

For first case $\epsilon =0.5$

There is need to find $N$ such that,

If $x>N$ then $\left|\frac{\sqrt{4x^2+1}}{x+1}-2\right|<0.5$

For the inequality,

  $-0.5<\frac{\sqrt{4x^2+1}}{x+1}-2<0.5$

Or $1.5<\frac{\sqrt{4x^2+1}}{x+1}<2.5$

Or $1.5<\frac{\sqrt{4x^2+1}}{x+1}<0.5$

One need to determine the values of $x$ for which given curve lies in between horizontal lines $y=1.5$ and $y=2.5$ .

Here cursor is use to estimate that curve crossing the line $y=1.5$ when $x\approx 2.82$ .

For getting the right of the number the curve stays between the line $y=1.5$ and $y=2.5$ .

Therefore after round the value of $x$ ,

If $x>3$ then $\left|\frac{\sqrt{4x^2+1}}{x+1}-2\right|<0.5$

Thus for $\epsilon =0.5$ .

One can select $N=3$ .

Graph of function $f\left(x\right)$ is:

  

For second case $\epsilon =0.1$ .

There is need to find $N$ such that

If $x>N$ then $\left|\frac{\sqrt{4x^2+1}}{x+1}-2\right|<0.1$

For the inequality,

  $-0.1<\frac{\sqrt{4x^2+1}}{x+1}-2<0.1$

Or $1.9<\frac{\sqrt{4x^2+1}}{x+1}<2.1$

One need to determine the values of $x$ for which given curve lies in between horizontal lines $y=1.9$ and $y=2.1$ .

Here cursor is use to estimate that curve crossing the line $y=1.9$ when $x\approx 18.87$ .

For getting the right of the number the curve stays between the line $y=1.9$ and $y=2.1$ .

Therefore after round the value of $x$ ,

If $x>19$ then $\left|\frac{\sqrt{4x^2+1}}{x+1}-2\right|<0.1$

Thus for $\epsilon =0.1$ .

We can select $N=19$ .

Graph of function $f\left(x\right)$ is: