8. $lim_{x\rightarrow0}\frac{e^{x}-1}{x}$\n|$x$|$-0.1$|$-0.01$|$-0.001$|$0.001$|$0.01$|$0.1$|\n|$f(x)$| | |…

Explanation:

Step1: Calculate for $x = - 0.1$ in $\lim_{x\rightarrow0}\frac{e^{x}-1}{x}$

Substitute $x=-0.1$ into $\frac{e^{x}-1}{x}$, we get $\frac{e^{-0.1}-1}{-0.1}=\frac{\frac{1}{e^{0.1}} - 1}{-0.1}\approx\frac{0.9048 - 1}{-0.1}=\frac{- 0.0952}{-0.1}=0.952$

Step2: Calculate for $x = - 0.01$ in $\lim_{x\rightarrow0}\frac{e^{x}-1}{x}$

Substitute $x = - 0.01$ into $\frac{e^{x}-1}{x}$, we get $\frac{e^{-0.01}-1}{-0.01}=\frac{\frac{1}{e^{0.01}}-1}{-0.01}\approx\frac{0.99005 - 1}{-0.01}=\frac{-0.00995}{-0.01}=0.995$

Step3: Calculate for $x=-0.001$ in $\lim_{x\rightarrow0}\frac{e^{x}-1}{x}$

Substitute $x=-0.001$ into $\frac{e^{x}-1}{x}$, we get $\frac{e^{-0.001}-1}{-0.001}=\frac{\frac{1}{e^{0.001}}-1}{-0.001}\approx\frac{0.9990005 - 1}{-0.001}=\frac{- 0.0009995}{-0.001}=0.9995$

Step4: Calculate for $x = 0.001$ in $\lim_{x\rightarrow0}\frac{e^{x}-1}{x}$

Substitute $x = 0.001$ into $\frac{e^{x}-1}{x}$, we get $\frac{e^{0.001}-1}{0.001}\approx\frac{1.0010005 - 1}{0.001}=\frac{0.0010005}{0.001}=1.0005$

Step5: Calculate for $x = 0.01$ in $\lim_{x\rightarrow0}\frac{e^{x}-1}{x}$

Substitute $x = 0.01$ into $\frac{e^{x}-1}{x}$, we get $\frac{e^{0.01}-1}{0.01}\approx\frac{1.01005 - 1}{0.01}=\frac{0.01005}{0.01}=1.005$

Step6: Calculate for $x = 0.1$ in $\lim_{x\rightarrow0}\frac{e^{x}-1}{x}$

Substitute $x = 0.1$ into $\frac{e^{x}-1}{x}$, we get $\frac{e^{0.1}-1}{0.1}\approx\frac{1.1052 - 1}{0.1}=\frac{0.1052}{0.1}=1.052$

For $\lim_{x\rightarrow0}\frac{\sin x}{x}$:

Step7: Calculate for $x = 1$

Substitute $x = 1$ into $\frac{\sin x}{x}$, we get $\frac{\sin1}{1}\approx0.8415$

Step8: Calculate for $x=-1$

Substitute $x=-1$ into $\frac{\sin x}{x}$, we get $\frac{\sin(-1)}{-1}=\frac{-\sin1}{-1}\approx0.8415$

Step9: Calculate for $x = 0.5$

Substitute $x = 0.5$ into $\frac{\sin x}{x}$, we get $\frac{\sin0.5}{0.5}\approx\frac{0.4794}{0.5}=0.9589$

Step10: Calculate for $x=-0.5$

Substitute $x=-0.5$ into $\frac{\sin x}{x}$, we get $\frac{\sin(-0.5)}{-0.5}=\frac{-\sin0.5}{-0.5}\approx0.9589$

Step11: Calculate for $x = 0.1$

Substitute $x = 0.1$ into $\frac{\sin x}{x}$, we get $\frac{\sin0.1}{0.1}\approx\frac{0.0998}{0.1}=0.9983$

Step12: Calculate for $x=-0.1$

Substitute $x=-0.1$ into $\frac{\sin x}{x}$, we get $\frac{\sin(-0.1)}{-0.1}=\frac{-\sin0.1}{-0.1}\approx0.9983$

Step13: Calculate for $x = 0.05$

Substitute $x = 0.05$ into $\frac{\sin x}{x}$, we get $\frac{\sin0.05}{0.05}\approx\frac{0.0499}{0.05}=0.9983$

Step14: Calculate for $x=-0.05$

Substitute $x=-0.05$ into $\frac{\sin x}{x}$, we get $\frac{\sin(-0.05)}{-0.05}=\frac{-\sin0.05}{-0.05}\approx0.9983$

Step15: Calculate for $x = 0.01$

Substitute $x = 0.01$ into $\frac{\sin x}{x}$, we get $\frac{\sin0.01}{0.01}\approx\frac{0.0099998}{0.01}=0.99998$

Step16: Calculate for $x=-0.01$

Substitute $x=-0.01$ into $\frac{\sin x}{x}$, we get $\frac{\sin(-0.01)}{-0.01}=\frac{-\sin0.01}{-0.01}\approx0.99998$

For $\lim_{x\rightarrow0^{+}}x\ln x$:

Step17: Calculate for $x = 0.1$

Substitute $x = 0.1$ into $x\ln x$, we get $0.1\times\ln(0.1)=0.1\times(- 2.3026)=-0.2303$

Step18: Calculate for $x = 0.01$

Substitute $x = 0.01$ into $x\ln x$, we get $0.01\times\ln(0.01)=0.01\times(-4.6052)=-0.0461$

Step19: Calculate for $x = 0.001$

Substitute $x = 0.001$ into $x\ln x$, we get $0.001\times\ln(0.001)=0.001\times(-6.9078)=-0.0069$

Step20: Calculate for $x = 0.0001$

Substitute $x = 0.0001$ into $x\ln x$, we get $0.0001\times\ln(0.0001)=0.0001\times(-9.2103)=-0.0009$

Step21: Calculate for $x = 0.00001$

Substitute $x = 0.00001$ into $x\ln x$, we get $0.00001\times\ln(0.00001)=0.00001\times(-11.5129)=-0.0001$

Answer:

$x$	-0.1	-0.01	-0.001	0.001	0.01	0.1
$f(x)$ for $\lim_{x\rightarrow0}\frac{e^{x}-1}{x}$	0.952	0.995	0.9995	1.0005	1.005	1.052

$x$	$\pm1$	$\pm0.5$	$\pm0.1$	$\pm0.05$	$\pm0.01$
$f(x)$ for $\lim_{x\rightarrow0}\frac{\sin x}{x}$	0.8415	0.9589	0.9983	0.9983	0.99998

$x$	0.1	0.01	0.001	0.0001	0.00001
$f(x)$ for $\lim_{x\rightarrow0^{+}}x\ln x$	-0.2303	-0.0461	-0.0069	-0.0009	-0.0001