) Use the derivative of sin (1/x) to show the sequence is decreasing.

Answer 1

Determine the derivative

f(x) = sin(1/x)

f'(x) = (-1/x²)cos(1/x)

Find critical values

0 = (-1/x²)cos(1/x)

0 = cos(1/x)

π/2 = 1/x

2/π = x

Use test points

f'(0.5) = (-1/0.5²)cos(1/0.5) = -4cos(2) = 1.664 > 0

f'(1) = (-1/1²)cos(1/1) = -cos(1) = -0.54 < 0

As the function's derivative decreases between these two test values, it is clear that the sequence f(x)=sin(1/x) is decreasing. Hope this helped!