1.

Examine the continuity of the funcation `f(x)={{: ((|sinx|)/x",", xne0),(1",",x=0 " at " x=0):}`

Answer» we have f(0) =1
`lim_(xto0+)f(x)=lim_(hto0)f(0+h)`
`lim_(hto0)(|sin(0 +h)|)/(( 0+h))=lim_(h to 0) (|sin h|)/(h) = lim_(h to0) (sinh)/h=1`
`lim_(xto0-)f(x)=lim_(hto0) f(0-h)`
` lim_(h to0) (|sin (-h)|)/ (-h) =lim_(hto0) (|-sin h|)/(-h)=lim_(hto0) (sinh)/(-h)=-1`
`lim_(xto0+) f(x)ne lim_(xto0-) f(x) . " so" , lim_(xto0) f(x)` does not exist.
Hence, f(x) is discontinuous at x=0.


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