f(x) = tan(1/x)

Calculus
Section 2.2

 

a.

tan(1/x)=0

general solution is

1/x=tan^-1(0)...........tan^-1(0)=0 (result in radians) or 0° (degrees)
periodicity of tan is π

1/x=0+π*n
1=0*x+x*π*n
x*π*n=1
x=1/(π*n).....where n=1,2,3,4,...

x=1/pi*1=1/π
x=1/pi*2=1/(2π)
x=1/pi*3= 1/(3π)

b.

tan(1/x)=1

general solution is

1/x=tan^-1(1)...........tan^-1(1)=π/4(result in radians)
periodicity of tan is π


1/x=π/4........cross multiply
4*1=x*π
x=4/π

general solution is

x=4/π+π*n =4/(π+4π*n)=4/(π(1+4n)) where n=0,1,2,3,4,...

x==4/(π(1+4*0)) =4/π

x=4/(π(1+4)) =4/(5π)

x=4/(π(1+4*2)) =4/(9π)

c.

 

This one is challenging. Good study notes for me.