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Examine the continuity of the following ...

Examine the continuity of the following functions at indicated points.
`f(x)={((sin2x)/x if xne0` at x=0),(2 if x=0):}`

Text Solution

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`lim_(xto0)f(x)=lim_(xto0)(sin2x)/x`
`lim_(xto0)2 cdot (sin2x)/(2x)=2 cdot lim_(2xto0)(sin2x)/(2x)=2 cdot 1`
`=2(becauselim_(theta=0)(sintheta)/theta=1)`
Again `f(0)=2implieslim_(xto0)f(x)=f(0)`
`therefore` The function f(x) is continuous at x=0
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