Question

Use your calculator and a table of values to find the exact value of limit as x goes to infinity of the product of x and the sine of 1 over x. (The limit as x approaches infinity)

Answer 1

check the picture below.

to do it using a spreadsheet, so... you simply enter a formula on the "y" column, in this case I used = sin(1/A1) <---- notice A1 is relative addressing.

from there, you can just drag the little black square on the cell, the cell handle and drag it as far as you want, it'll copy the formula as you go.

Answer 2

The exact value is 1.

Explanation:

Here, given expression,

`x\sin((1)/(x))`

When we put, x = 5, 10, 15, 20, 25, 30, 35, 40, 45, 55,.....

In the expression,

We found that the value of the expression is approaches to 1,

Hence, the exact value of,

`\lim_(x\rightarrow \infty) x\sin((1)/(x))` is 1.

Alternative method :

`\lim_(x\rightarrow \infty) x\sin((1)/(x))`

`=\lim_(x\rightarrow \infty) (\sin((1)/(x)))/((1)/(x))`

By L'hospital's rule,

`=\lim_(x\rightarrow \infty)((-\cos((1)/(x)))/(x^2))/(-(1)/(x^2))`

`=\lim_(x\rightarrow \infty) (\cos((1)/(x)))/(1)`

`=\lim_(x\rightarrow \infty) \cos((1)/(x))`

`=\cos((1)/(\infty))`

`=\cos(0)`

`=1`