Question

Guess the value of the limit

\(\mathop {{\bf{lim}}}\limits_{x \to - \infty } \frac{{{x^{\bf{2}}}}}{{{{\bf{2}}^x}}}\)

by evaluating the function \(f\left( x \right) = \frac{{{x^{\bf{2}}}}}{{{{\bf{2}}^x}}}\) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 50, and 100. Then use a graph of f to support your guess.

Short answer

The value of the expression \(\mathop {\lim }\limits_{x \to \infty } \frac{{{x^2}}}{{{2^x}}}\) is \(0\).

The graph is shown below:

The function\(f\left( x \right)\) also suggests that \(f\left( x \right)\) is approaching 0 as x is approaching to \(\infty \).

Step-by-step solution

Step 1:Find the values of \(f\left( x \right)\)

Obtain the value of \(f\left( 0 \right)\) as:

\(\begin{array}{c}f\left( 0 \right) = \frac{{{0^2}}}{{{2^0}}}\\ = 0\end{array}\)

Obtain the value of \(f\left( 1 \right)\) as:

\(\begin{array}{c}f\left( 1 \right) = \frac{{{1^2}}}{{{2^1}}}\\ = \frac{1}{2}\end{array}\)

The table below represents the values of \(f\left( x \right)\), calculated by using the calculator.

\(x\)	\(f\left( x \right) = \frac{{{x^{\bf{2}}}}}{{{{\bf{2}}^x}}}\)
0	0
1	0.5
2	1
3	1.125
4	1
5	0.78125
6	0.5625
7	0.3828125
8	0.25
9	0.158203125
10	0.09765625
20	0.00038147
50	\(2.2204 \times {10^{ - 12}}\)
100	\(7.8886 \times {10^{ - 27}}\)

From the table, it can be observed that the value of the expression is \(\mathop {\lim }\limits_{x \to \infty } \frac{{{x^2}}}{{{2^x}}} = 0\).

Sketch the graph of \(f\left( x \right)\)

Use the following steps to plot the graph of given functions:

1. In the graphing calculator, select “STAT PLOT” and enter the equation \({X^2}/{2^X}\) in the \({Y_1}\) tab.
2. Enter the graph button in the graphing calculator.

The figure below represents the sketch of the function.