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Prove that the function f(x)= {:{((sinx...

Prove that the function ` f(x)= {:{((sinx)/x , xlt 0), (x^(2)+1, x ge 0):}`
in x belongs to R is continuous

Text Solution

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Hint : Let ' a' be any arbitrary in the damain of f(x)
Case I : a lt 0 , f is continuous in `(-oo,0)`
`lim_(x to a) (sinx)/x = sina/a = f(a)`
Case II : a gt 0 , is continuous in ` (0 ,oo)`
`lim_(x to a) f(x)=lim_(x to a) (x^(2)+1) =a^(2)+1= f(a)`
Case III : a = 0,
Left hand limit ` lim_( x to 0^(-)) f(x)-1`
Right hand limit ` lim_(x to 0+) f(x)=1`
The function is continuous in R.
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