Question

Verify whether the function f defined by f(x) = `{(x sin (1/x)",", x ≠ 0), (0",", x = 0):}`  is continuous at x = 0 or not. 

Answer 1

Given, f(x) = `{(x sin (1/x)",", x ≠ 0), (0",", x = 0):}`

for a continuous function, LHL = RHL = f(a)

Now, LHL = `lim_(x->0^-) x sin  1/x`

= `lim_(h->0) (0 - h) sin (1/ (0 - h))`

= `lim_(h->0) - h sin (-1/h)`

= `lim_(h -> 0) h sin (1/h)`     ...[sin (−∞) = − sin θ]

= 0 × sin (∞)

= 0

and RHL = `lim_(x->0^+) x sin  1/x`

= `lim_(h->0) (0 + h) sin (1/(0 + h))`

= `lim_(h->0) h sin (1/h)`

= 0.sin(∞) = 0

So, LHL = RHL = f(0)

Hence, f(x) is continuous at x = 0