Question

Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically.

\(\mathop {\lim }\limits_{x \to {0^ + }} {x^x}\)

Short answer

The value of the limit is \(\mathop {\lim }\limits_{x \to {0^ + }} f\left( x \right) = 1\).

Step-by-step solution

Intuitive Definition of a Limit

Let \(f\left( x \right)\) be defined when \(x\) is near the number \(a\). (Which means that \(f\) is defined on some open interval that contains \(a\), except possibly at \(a\) itself.) Write it as \(\mathop {\lim }\limits_{x \to a} f\left( x \right) = L\).

The limit of \(f\left( x \right)\) as \(x\) approaches \(a\) is equal to \(L\). When the values of \(f\left( x \right)\) arbitrarily close to \(L\) by restricting \(x\) to be sufficiently close to \(a\) but not equal to\(a\).

Compare the definition 1 and 2, we obtain the following is true.

\(\mathop {\lim }\limits_{x \to a} f\left( x \right) = L\) if and only if \(\mathop {\lim }\limits_{x \to {a^ - }} f\left( x \right) = L\) and \(\mathop {\lim }\limits_{x \to {a^ + }} f\left( x \right) = L\).

Use a table to estimate the value of a limit

Let \(f\left( x \right) = {x^x}\).

Evaluate the function at \(t = 0.1\) as shown below:

\(\begin{aligned}\mathop {\lim }\limits_{x \to {{0.1}^ + }} {x^x} &= {\left( {0.1} \right)^{0.1}}\\ &= 0.794328\end{aligned}\)

The other values are obtained in a similar way listed in the table as shown below:

\(x\)	\(f\left( x \right)\)
\(\begin{aligned}0.1\\0.01\\0.001\\0.0001\end{aligned}\)	\(\begin{aligned}0.794328\\0.954993\\0.993116\\0.999079\end{aligned}\)

It is observed from the table that left, and right limits are the same and therefore, \(\mathop {\lim }\limits_{x \to {0^ + }} f\left( x \right) = 1\).

The procedure to draw the graph of the equation by using the graphing calculator is as follows:

To check the answer visually, draws the graph of the function\(f\left( x \right) = {x^x}\)by using the graphing calculator as shown below:

1. Open the graphing calculator. Select the “STAT PLOT” and enter the equation\({X^X}\)in the\({Y_1}\)tab.
2. Enter the “GRAPH” button in the graphing calculator.

Visualization of the graph of the function \(f\left( x \right) = {x^x}\)is shown below:

It is observed from the graph that the result of the limit is 1 is confirmed.

Thus, the value of the limit is \(\mathop {\lim }\limits_{x \to {0^ + }} f\left( x \right) = 1\).