Chapter 8.2, Problem 70E

a.

To determine

To state: The reason why some graphs of $f$ may give false information about ${}_x{\underrightarrow{\lim }}_{0}f\left(x\right)$ . Try the window $\left[-0.3,0.3\right]\text{ by }\left[-0.5,1\right]$ .

Answer to Problem 70E

The reason is the minor difference in the numerator is and the value being subtracted.

Explanation of Solution

Given information:

The given statement says to state the reason why some graphs of $f$ may give false information about ${}_x{\underrightarrow{\lim }}_{0}f\left(x\right)$ .

The reason about why some graphs of $f$ may give false information about ${}_x{\underrightarrow{\lim }}_{0}f\left(x\right)$ , because the difference in the numerator is so small compared to the value being subtracted, any calculator or computer with limited precision will give the incorrect result that $1-\cos x^6\text{ is 0}$ for even moderately small value of x .

For example at $x=0.1$ the value of $\cos x^6=0.9999999999995$ (13 places) but on a 10 place calculator, the values are $\cos x^6=1\text{ and }\cos x^6=0$ .

Therefore, the ultimate reason is that the difference in the numerator is so small when compared to the value being subtracted.

b.

To determine

To state: The reason why tables may give false information about ${}_x{\underrightarrow{\lim }}_{0}f\left(x\right)$ .

Answer to Problem 70E

The reason is the minor difference in the numerator is and the value being subtracted.

Explanation of Solution

Given information:

The given statement says to reason why tables may give false information about ${}_x{\underrightarrow{\lim }}_{0}f\left(x\right)$ .

The reason about why the table may give false information about ${}_x{\underrightarrow{\lim }}_{0}f\left(x\right)$ , because the difference in the numerator is so small compared to the value being subtracted, any calculator or computer with limited precision will give the incorrect result that $1-\cos x^6\text{ is 0}$ for even moderately small value of x .

Therefore, the ultimate reason is that the difference in the numerator is so small when compared to the value being subtracted.

c.

To determine

To state: The value of ${}_x{\underrightarrow{\lim }}_{0}f\left(x\right)$ using the $\text{l’Hospital’s}$ Rule.

Answer to Problem 70E

The resultant answer is ${}_x{\underrightarrow{\lim }}_{0}f\left(x\right)=\frac{1}{2}$ .

Explanation of Solution

Given information:

The given information is ${}_x{\underrightarrow{\lim }}_{0}f\left(x\right)$ .

Find the value of ${}_x{\underrightarrow{\lim }}_{0}f\left(x\right)$ ,

  $\begin{array}{c}{}_x{\underrightarrow{\lim }}_{0}f\left(x\right){=}_x{\underrightarrow{\lim }}_0\frac{1-\cos x^6}{x^{12}}\\ =\frac{1-\cos 0^6}{0^{12}}\\ =\frac{1-\cos 0}{0}\\ =\frac{0}{0}\end{array}$

It can be seen that substituting $x=0$ leads to the indeterminate form $\frac{0}{0}$ because the numerator and denominator of the fraction are 0 when 0 is substituted for x . So apply the $\text{l’Hospital’s}$ Rule. Differentiate numerator and denominator.

  $\begin{array}{c}{}_x{\underrightarrow{\lim }}_0\frac{1-\cos x^6}{x^{12}}{=}_x{\underrightarrow{\lim }}_0\frac{-6x^5\left(-\sin x^6\right)}{12x^{11}}\\ {=}_x{\underrightarrow{\lim }}_0\frac{\sin x^6}{2x^6}\\ =\frac{\sin 0^6}{2{\left(0\right)}^6}\\ =\frac{0}{0}\end{array}$

It can be seen that substituting $x=0$ leads to 0 in both the numerator and denominator of the second fraction, so differentiate again,

  $\begin{array}{c}{}_x{\underrightarrow{\lim }}_0\frac{\sin x^6}{2x^6}{=}_x{\underrightarrow{\lim }}_0\frac{6x^5\left(\cos x^6\right)}{12x^5}\\ {=}_x{\underrightarrow{\lim }}_0\frac{\cos x^6}{2}\\ =\frac{\cos 0^6}{2}\\ =\frac{1}{2}\end{array}$

Therefore, the value is ${}_x{\underrightarrow{\lim }}_{0}f\left(x\right)=\frac{1}{2}$ .

d.

To determine

To state: The statement in one’s own words “this is an example of a function for which $\text{graphers}$ do not have enough precision to give reliable information.

Answer to Problem 70E

Because of the round-off error in computing values of this function on a limited precision device, $\text{graphers}$ do not have enough precision to give reliable information.

Explanation of Solution

Given information:

The given statement says to write to the statement in one’s own words “this is an example of a function for which $\text{graphers}$ do not have enough precision to give reliable information.

The graph and/or table of a graph show the value of the function to be 0 for x - value moderately close to 0, but the limit is $\frac{1}{2}$ .

The calculator is giving unreliable information because there is significant round-off error in computing values of this function on a limited precision device.

Therefore, $\text{graphers}$ do not have enough precision to give reliable information because of the round-off error in computing values of this function on a limited precision device.