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A function f(x) is defined as follows :f(x)={(x sin ""1/x "," x ne 0),(0 "," x=0):} Discuss its continuity at x=0 |
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Answer» Solution :At x=0 f(0)=0 R.H.L `=UNDERSET(xrarr0^+)limf(x)=underset(hrarr0)limf(0+h)` `=underset(hrarr0)limhsin"" (1)/(h)` `=0xx`(a definite value ) =0 `(therefore)"SIN" (1)/(h) ` ALWAYS lies between -1 and 1) gt L.H.L`=underset(xrarr0^-)lim f(x)=underset(hrarr0)limf(0-h)` ` =underset(hrarr0)lim(-h)sin (1/-h)=underset(hrarr0)(lim) h sin ""1/h` `=0xx` (a definite value )=0 `therefore`R.H.L = f(0)=L.H.L `therefore`f(x) is continuous at x=0. Prove that(x) is CONTINOUS at x=2. |
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