Question

numerical, graphical, and analytic analysis in exercises 47 - 50, use a graphing utility to complete the table and estimate the limit as x approaches infinity. then use a graphing utility to graph the function and estimate the limit. finally, find the limit analytically and compare your results with the estimates.
\begin{tabular}{|c|c|c|c|c|c|c|c|}hline$x$&$10^{0}$&$10^{1}$&$10^{2}$&$10^{3}$&$10^{4}$&$10^{5}$&$10^{6}$\hline$f(x)$& & & & & & & \hlineend{tabular}

1. $f(x)=x - sqrt{x(x - 1)}$
2. $f(x)=x^{2}-xsqrt{x(x - 1)}$
3. $f(x)=xsin\frac{1}{2x}$
4. $f(x)=\frac{x + 1}{xsqrt{x}}$

Explanation:

Step1: Consider the function $f(x)=x - \sqrt{x(x - 1)}$

Rationalize the function. Multiply and divide by the conjugate $x+\sqrt{x(x - 1)}$.
\[

$$\begin{align*} f(x)&=\frac{(x - \sqrt{x(x - 1)})(x+\sqrt{x(x - 1)})}{x+\sqrt{x(x - 1)}}\\ &=\frac{x^{2}-x(x - 1)}{x+\sqrt{x(x - 1)}}\\ &=\frac{x^{2}-x^{2}+x}{x+\sqrt{x(x - 1)}}\\ &=\frac{x}{x+\sqrt{x^{2}-x}} \end{align*}$$

\]

Step2: Divide numerator and denominator by $x$

For $x>0$, we have $\frac{x}{x+\sqrt{x^{2}-x}}=\frac{1}{1+\sqrt{1-\frac{1}{x}}}$

Step3: Find the limit as $x\to\infty$

As $x\to\infty$, $\frac{1}{x}\to0$. So, $\lim_{x\to\infty}\frac{1}{1+\sqrt{1-\frac{1}{x}}}=\frac{1}{1 + 1}=\frac{1}{2}$

To complete the table for $x = 10^{0},10^{1},10^{2},10^{3},10^{4},10^{5},10^{6}$:

• When $x = 10^{0}=1$, $f(1)=1-\sqrt{1\times(1 - 1)}=1$
• When $x = 10^{1}=10$, $f(10)=10-\sqrt{10\times(10 - 1)}=10 - \sqrt{90}\approx10-9.49 = 0.51$
• When $x = 10^{2}=100$, $f(100)=100-\sqrt{100\times(100 - 1)}=100-\sqrt{9900}\approx100 - 99.5=0.5$
• When $x = 10^{3}=1000$, $f(1000)=1000-\sqrt{1000\times(1000 - 1)}=1000-\sqrt{999000}\approx1000 - 999.5=0.5$
• When $x = 10^{4}=10000$, $f(10000)=10000-\sqrt{10000\times(10000 - 1)}=10000-\sqrt{99990000}\approx10000 - 9999.5=0.5$
• When $x = 10^{5}=100000$, $f(100000)=100000-\sqrt{100000\times(100000 - 1)}=100000-\sqrt{9999900000}\approx100000 - 99999.5=0.5$
• When $x = 10^{6}=1000000$, $f(1000000)=1000000-\sqrt{1000000\times(1000000 - 1)}=1000000-\sqrt{999999000000}\approx1000000 - 999999.5=0.5$

Answer:

The limit of $f(x)=x - \sqrt{x(x - 1)}$ as $x\to\infty$ is $\frac{1}{2}$. The values in the table for increasing - large values of $x$ approach $0.5$, which is consistent with the analytical result.