Question

Discuss the continuity of the function f(x)=xtan(1/x) at x = 0 given that x≠0 and f(0)=0. please solve it

Answer 1

f(x)=xtan(1/x)

at x=0 , f(0)=0

LHL=lim f(x)

x- 0*-

=lim f(0-h)

h-0*-

=lim (0-h)tan(1/0-h)

h-0*

= lim -h(-tan(1/h))

h-0

= lim h tan(1/h)

h-0

=. 0

RHL = lim f(x)

x-0*+

=lim f(0+h)

h-0

=lim (0+h) tan(1/(0+h)

h-0

=lim h tan (1/h)

h-0

=0

since f(0)=LHL=RHL

Hence the function is continuous at x=0