Chapter 2.5, Problem 11E

To determine

To evaluate: The value of $\underset{x\to \infty }{\lim }\frac{x^2}{2^x}$ for the function $f\left(x\right)=\frac{x^2}{2^x}$ for the values

$x=0,1,2,3,4,5,6,7,8,9,10,20,50\text{ and 100}$.

Answer to Problem 11E

The value of $\underset{x\to \infty }{\lim }\frac{x^2}{2^x}$ by evaluating the function $f\left(x\right)=\frac{x^2}{2^x}$ for

$x=0,1,2,3,4,5,6,7,8,9,10,20,50\text{ and 100}$ is 0.

Explanation of Solution

Calculation:

Consider the function $f\left(x\right)=\frac{x^2}{2^x}$

The table value of the limit function $f\left(x\right)$.

x	$x^2$	$2^x$	$f\left(x\right)$
0	0	1	0
1	1	2	0.5
2	4	4	1
3	9	8	1.125
4	16	16	1
5	25	32	0.78125
6	36	64	0.5625
7	49	128	0.3828125
8	64	256	0.25
9	81	512	0.15820313
10	100	1024	0.09765625
20	400	1048576	3.814697$\times {10}^{-4}$
50	2500	$1.125899\times {10}^{15}$	2.220446$\times {10}^{-12}$
100	10000	$1.267650\times {10}^{30}$	0

From the table, it is concluded that $\underset{x\to \infty }{\lim }\frac{x^2}{2^x}=0$.

Graph:

Use the online graphing calculator to plot the graph of the function $f\left(x\right)=\frac{x^2}{2^x}$ as shown in Figure1.

From Figure1, it is observed that the curve approaches 0 as x approaches to infinity.

From the above graph, it is concluded that $\underset{x\to \infty }{\lim }\frac{x^2}{2^x}=0$.

Therefore, the graph supports the conclusion from the table values of f.