1.

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

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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