QuestionFind the limit as $x$ approaches infinity for $\frac{x^{2}}{2^{x}}$ . Evaluate $f(x)=\frac{x^{2}}{2^{x}}$ for $x=0,1,\ldots,100$ .

Studdy Solution

STEP 1

Assumptions1. We are asked to find the limit as x approaches infinity of the function $f(x)=\frac{x^{}}{^{x}}$ . . We are asked to evaluate the function at $x=0,1,,3,4,5,6,7,8,9,10,20,50$ , and100.
3. We are asked to use a graph of the function to support our guess.

STEP 2

Let's start by evaluating the function $f(x)=\frac{x^{2}}{2^{x}}$ at the given values of x.

STEP 3

Evaluate the function at $x=0$ .
$f(0) = \frac{0^{2}}{2^{0}} =0$

STEP 4

Evaluate the function at $x=1$ .
$f(1) = \frac{1^{2}}{2^{1}} =0.$

STEP 5

Evaluate the function at $x=2$ .
$f(2) = \frac{2^{2}}{2^{2}} =1$

STEP 6

Evaluate the function at $x=3$ .
$f(3) = \frac{3^{2}}{2^{3}} =1.125$

STEP 7

Evaluate the function at $x=4$ .
$f(4) = \frac{4^{2}}{2^{4}} =1$

STEP 8

Evaluate the function at $x=5$ .
$f(5) = \frac{5^{2}}{2^{5}} =0.78125$

STEP 9

Evaluate the function at $x=6$ .
$f(6) = \frac{6^{2}}{2^{6}} =.5625$

STEP 10

Evaluate the function at $x=7$ .
$f(7) = \frac{7^{2}}{2^{7}} =0.3828125$

STEP 11

Evaluate the function at $x=8$ .
$f(8) = \frac{8^{}}{^{8}} =0.25$

STEP 12

Evaluate the function at $x=9$ .
$f(9) = \frac{9^{2}}{2^{9}} =0.158203125$

STEP 13

Evaluate the function at $x=10$ .
$f(10) = \frac{10^{2}}{2^{10}} =0.09765625$

STEP 14

Evaluate the function at $x=20$ .
$f(20) = \frac{20^{2}}{2^{20}} =0.00009536743$

STEP 15

Evaluate the function at $x=50$ .
$f(50) = \frac{50^{2}}{2^{50}} \approx0$

STEP 16

Evaluate the function at $x=100$ .
$f(100) = \frac{100^{2}}{2^{100}} \approx0$

STEP 17

From the values obtained, we can see that as x increases, the value of the function decreases and approaches0.

STEP 18

To further support our guess, we can plot a graph of the function $f(x)=\frac{x^{2}}{2^{x}}$ .

STEP 19

From the graph, we can see that as x approaches infinity, the value of the function approaches.

STEP 20

Therefore, we can guess that the limit of the function as x approaches infinity is0.
$\lim{x \rightarrow \infty} \frac{x^{}}{^{x}} =0$

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