a function (f) and value (a) are given below.(f(x)=\frac{2}{x + 9}), (a = 5)approximate the limit of the…

a function (f) and value (a) are given below.(f(x)=\frac{2}{x + 9}), (a = 5)approximate the limit of the difference quotient,(lim_{h\rightarrow0}\frac{f(a + h)-f(a)}{h}),using (h=pm0.1,pm0.01). (round your answer to within three decimal places if necessary, but do not round until your final computation.)a. when (h = 0.1), (\frac{f(a + h)-f(a)}{h}=)b. when (h=-0.1), (\frac{f(a + h)-f(a)}{h}=)c. when (h = 0.01), (\frac{f(a + h)-f(a)}{h}=)d. when (h=-0.01), (\frac{f(a + h)-f(a)}{h}=)

a function (f) and value (a) are given below.(f(x)=\frac{2}{x + 9}), (a = 5)approximate the limit of the difference quotient,(lim_{h\rightarrow0}\frac{f(a + h)-f(a)}{h}),using (h=pm0.1,pm0.01). (round your answer to within three decimal places if necessary, but do not round until your final computation.)a. when (h = 0.1), (\frac{f(a + h)-f(a)}{h}=)b. when (h=-0.1), (\frac{f(a + h)-f(a)}{h}=)c. when (h = 0.01), (\frac{f(a + h)-f(a)}{h}=)d. when (h=-0.01), (\frac{f(a + h)-f(a)}{h}=)

Answer

Explanation:

Step1: Find (f(a))

Given (f(x)=\frac{2}{x + 9}) and (a = 5), then (f(a)=f(5)=\frac{2}{5+9}=\frac{2}{14}=\frac{1}{7})

Step2: General formula for (f(a + h))

(f(a + h)=\frac{2}{(a + h)+9}=\frac{2}{(5 + h)+9}=\frac{2}{h + 14})

Step3: Calculate (\frac{f(a + h)-f(a)}{h})

(\frac{f(a + h)-f(a)}{h}=\frac{\frac{2}{h + 14}-\frac{1}{7}}{h}=\frac{\frac{14-(h + 14)}{7(h + 14)}}{h}=\frac{\frac{-h}{7(h + 14)}}{h}=-\frac{1}{7(h + 14)})

Step4a: When (h = 0.1)

Substitute (h = 0.1) into (-\frac{1}{7(h + 14)}), we get (-\frac{1}{7(0.1+14)}=-\frac{1}{7\times14.1}=-\frac{1}{98.7}\approx - 0.010)

Step4b: When (h=-0.1)

Substitute (h=-0.1) into (-\frac{1}{7(h + 14)}), we get (-\frac{1}{7(-0.1 + 14)}=-\frac{1}{7\times13.9}=-\frac{1}{97.3}\approx - 0.010)

Step4c: When (h = 0.01)

Substitute (h = 0.01) into (-\frac{1}{7(h + 14)}), we get (-\frac{1}{7(0.01+14)}=-\frac{1}{7\times14.01}=-\frac{1}{98.07}\approx - 0.010)

Step4d: When (h=-0.01)

Substitute (h=-0.01) into (-\frac{1}{7(h + 14)}), we get (-\frac{1}{7(-0.01+14)}=-\frac{1}{7\times13.99}=-\frac{1}{97.93}\approx - 0.010)

Answer:

a. (-0.010) b. (-0.010) c. (-0.010) d. (-0.010)