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Question
Examine the continuity of the followin function :
`{:(,f(x),=x^2cos(1/x),",","for "x!=0),(,,=0,",","for "x=0):}}" at "x=0`
Solution
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
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