Question

(a) Evaluate \(h\left( x \right) = \frac{{\left( {\tan x - x} \right)}}{{{x^3}}}\)for \(x = {\bf{0}}.{\bf{5}},{\rm{ }}{\bf{0}}.{\bf{1}},{\rm{ }}{\bf{0}}.{\bf{05}},{\rm{ }}{\bf{0}}.{\bf{01}}\), and 0.005. (b) Guess the value of \(\mathop {\lim }\limits_{x \to 0} \frac{{\tan x - x}}{{{x^3}}}\).

(c) Evaluate \(h\left( x \right)\)for successively smaller values of xuntil you finally reach 0 values for \(h\left( x \right)\). Are you still confident that your guess in part (b) is correct? Explain why you eventually obtained 0 values. (In Section 3.7 a method for evaluating the limit will be explained.)

(d) Graph the function h in the viewing rectangle \(\left( { - 1,\;1} \right)\)by\(\left( {0,\;1} \right)\). Then zoom in toward the point where the graph crosses the y-axis to estimate the limit of \(h\left( x \right)\)as xapproaches 0. Continue to zoom in until you observe distortions in the graph of h. Compare with the results of part (c).

Short answer

(a) The values of \(h\left( x \right)\) for the given values of \(x\)are\(0.557408\),\(0.370420\), \(0.334672\),\(0.333667\), \(0.333347\), and \(0.333337\).

(b) The values of \(\mathop {\lim }\limits_{x \to 0} \frac{{\tan x - x}}{{{x^3}}}\)is\(0.33\).

(c) The guess is not correct, and the explanation is given by the behavior of the numerator and the capability of the calculator as \(x\) approaches to zero.

(d) The graph of the function \(h\) is as follows.

The limit \(\mathop {\lim }\limits_{x \to 0} h\left( x \right)\) does not exist, and in the part (c), \(\mathop {\lim }\limits_{x \to 0} h\left( x \right) = 0\), and in this part the limit does not exist.

Step-by-step solution

Describe the given information

The given function is as follows.

\(h\left( x \right) = \frac{{\left( {\tan x - x} \right)}}{{{x^3}}}\)

Find the value of the function for the given values of x

Substitute \(x = 0.5\) in function\(h\left( x \right) = \frac{{\left( {\tan x - x} \right)}}{{{x^3}}}\).

\(\begin{array}{c}h\left( x \right) = \frac{{\left( {\tan \left( 1 \right) - 1} \right)}}{{{{\left( 1 \right)}^3}}}\\ = 0.557408\end{array}\)

Similarly, calculate the other values of \(h\left( x \right)\) for the different values of \(x\) as shown in the following table.

\(x\)	\(h\left( x \right) = \frac{{\left( {\tan x - x} \right)}}{{{x^3}}}\)
\(1\)	\(0.557408\)
\(0.5\)	\(0.370420\)
\(0.1\)	\(0.334672\)
\(0.05\)	\(0.333667\)
\(0.01\)	\(0.333347\)
\(0.005\)	\(0.333337\)

Therefore, the values of \(h\left( x \right)\) for the given values of \(x\)are\(0.557408\),\(0.370420\),\(0.334672\),\(0.333667\),\(0.333347\), and\(0.333337\).

Guess the value of \(\mathop {\lim }\limits_{x \to 0} \frac{{\tan x - x}}{{{x^3}}}\).

From the table, it can be observed that as \(x\)tends to 0, the function approaches to 0.33.

Therefore, the values of \(\mathop {\lim }\limits_{x \to 0} \frac{{\tan x - x}}{{{x^3}}}\)is\(0.33\).

Find the value of the function for the successively smaller values of x, determine whether the guess is correct, and explain the reason behind obtaining the 0 value.

Calculate the other values of \(h\left( x \right)\) for the different values of \(x\) as shown in the following table.

\(x\)	\(h\left( x \right) = \frac{{\left( {\tan x - x} \right)}}{{{x^3}}}\)
\(1\)	\(0.557408\)
\(0.5\)	\(0.370420\)
\(0.1\)	\(0.334672\)
\(0.05\)	\(0.333667\)
\(0.01\)	\(0.333347\)
\(0.005\)	\(0.333337\)
\(0.001\)	\(0.333333\)
\(0.0001\)	\(0.333333\)
\(0.00001\)	\(0.333333\)
\(0.000001\)	\(0.333307\)
\(0.0000001\)	\(0.330872\)
\(0.00000001\)	\(0.000000\)

As \(x\) is approaches to zero, the value of \(\tan x\) and \(x\) become closer and there is a value where the calculator capability is reached, and the subtraction in the numerator is zero. Therefore, it cannot be said confidentially that the guess made in part (b) is correct. The explanation is given by the behavior of the numerator and the capability of the calculator as \(x\) approaches to zero.

Draw the graph of the function h in the viewing rectangle \(\left( { - 1,\;1} \right)\)by\(\left( {0,\;1} \right)\), estimate the limit of \(h\left( x \right)\) as x approaches 0 by zooming on the graph, and continue to zoom in to observe distortions in the graph of h. Compare with the results of part (c).

The graph of the function \(h\) is as follows.

Zoom in the graph to get the limit of\(h\left( x \right)\).

From the graph it can be observed that \(\mathop {\lim }\limits_{x \to 0} h\left( x \right)\) does not approach to a specific value, then the limit does not exist.

Zoom in the graph to get the distorted graph.

In part (c), \(\mathop {\lim }\limits_{x \to 0} h\left( x \right) = 0\), and in this part the limit does not exist.