Question

Examine the continuity of the followin function : 

  `{:(,f(x),=x^2cos(1/x),",","for "x!=0),(,,=0,",","for "x=0):}}" at "x=0`   

Answer 1

f (x) = `"x"^2  "cos" (1/"x")` , for x ≠ 0

       = 0 ,      for  x = 0

∴ f(0) = 0 

We know that ∀ x ∈ R

cos `(1/"x") in` [-1 , 1] i.e. finite number = k (say)

`therefore lim_(x -> 0)  "x"^2  "cos"  (1/"x"^2) = lim_(x -> 0)  "x"^2 . "k"` 

where k ∈ [-1 , 1]

= 0

`therefore lim_(x -> 0) "f(x)" = "f"(0) = 0`  

Hence function is continuous at x = 0