numerical, graphical, and analytic analysis in exercises 47 - 50, use a graphing utility to complete the…

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.\n\n| x | 10^0 | 10^1 | 10^2 | 10^3 | 10^4 | 10^5 | 10^6 |\n|----|----|----|----|----|----|----|----|\n| f(x) | | | | | | | |\n\n47. f(x)=x - \\sqrt{x(x - 1)}\n48. f(x)=x^2 - x\\sqrt{x(x - 1)}\n49. f(x)=x\\sin\\frac{1}{2x}\n50. f(x)=\\frac{x + 1}{x\\sqrt{x}}

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.\n\n| x | 10^0 | 10^1 | 10^2 | 10^3 | 10^4 | 10^5 | 10^6 |\n|----|----|----|----|----|----|----|----|\n| f(x) | | | | | | | |\n\n47. f(x)=x - \\sqrt{x(x - 1)}\n48. f(x)=x^2 - x\\sqrt{x(x - 1)}\n49. f(x)=x\\sin\\frac{1}{2x}\n50. f(x)=\\frac{x + 1}{x\\sqrt{x}}

Answer

Explanation:

Step1: Rationalize the function for 47.

For (f(x)=x - \sqrt{x(x - 1)}), multiply and divide by (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*} ] Divide numerator and denominator by (x) (since (x>0) as (x\to+\infty)): [ \begin{align*} f(x)&=\frac{1}{1+\sqrt{1-\frac{1}{x}}} \end{align*} ] As (x\to+\infty), (\frac{1}{x}\to0), so (\lim_{x\to+\infty}f(x)=\frac{1}{2})

To complete the table for (x = 10^{n},n = 0,1,\cdots,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\times9}=10 - 3\sqrt{10}\approx10-9.49 = 0.51) When (x = 10^{2}=100), (f(100)=100-\sqrt{100\times99}=100 - 10\sqrt{99}\approx100 - 99.5=0.5) When (x = 10^{3}=1000), (f(1000)=1000-\sqrt{1000\times999}=1000 - 10\sqrt{99900}\approx1000 - 999.5 = 0.5) As (x) gets larger, (f(x)) approaches (0.5)

Step2: Analyze function 48.

For (f(x)=x^{2}-x\sqrt{x(x - 1)}), factor out (x^{2}): [ \begin{align*} f(x)&=x^{2}(1-\sqrt{1-\frac{1}{x}}) \end{align*} ] Multiply and divide (1-\sqrt{1 - \frac{1}{x}}) by (1+\sqrt{1-\frac{1}{x}}): [ \begin{align*} 1-\sqrt{1-\frac{1}{x}}&=\frac{(1-\sqrt{1-\frac{1}{x}})(1 + \sqrt{1-\frac{1}{x}})}{1+\sqrt{1-\frac{1}{x}}}\ &=\frac{1-(1-\frac{1}{x})}{1+\sqrt{1-\frac{1}{x}}}\ &=\frac{\frac{1}{x}}{1+\sqrt{1-\frac{1}{x}}} \end{align*} ] So (f(x)=\frac{x}{1+\sqrt{1-\frac{1}{x}}}), and (\lim_{x\to+\infty}f(x)=+\infty)

Step3: Analyze function 49.

For (f(x)=x\sin\frac{1}{2x}), let (t=\frac{1}{x}), then as (x\to+\infty), (t\to0) [ \begin{align*} f(x)&=\frac{\sin\frac{t}{2}}{t} \end{align*} ] We know that (\lim_{u\to0}\frac{\sin u}{u}=1), so (\lim_{t\to0}\frac{\sin\frac{t}{2}}{t}=\frac{1}{2}\lim_{t\to0}\frac{\sin\frac{t}{2}}{\frac{t}{2}}=\frac{1}{2})

Step4: Analyze function 50.

For (f(x)=\frac{x + 1}{x\sqrt{x}}=\frac{1}{x^{\frac{1}{2}}}+\frac{1}{x^{\frac{3}{2}}}) As (x\to+\infty), (\lim_{x\to+\infty}\frac{1}{x^{\frac{1}{2}}}=0) and (\lim_{x\to+\infty}\frac{1}{x^{\frac{3}{2}}}=0), so (\lim_{x\to+\infty}f(x)=0)

Answer:

For 47: (\lim_{x\to+\infty}(x-\sqrt{x(x - 1)})=\frac{1}{2}) For 48: (\lim_{x\to+\infty}(x^{2}-x\sqrt{x(x - 1)})=+\infty) For 49: (\lim_{x\to+\infty}(x\sin\frac{1}{2x})=\frac{1}{2}) For 50: (\lim_{x\to+\infty}\frac{x + 1}{x\sqrt{x}}=0)