Discuss of the continutiy of the fuction `f(x)={{:(sin x/2 ", " x lt 0),(x+2 "," x ge 0):}`

Explanation

When `x lt 0`
`f(x) =(sin x)/(x)`
We know that sin x and x both are everywhere continuous .
`therefore (sin x)/(x)` is also continuous when `x lt 0`.
When ` x gt 0`
f(x) =x+2 is a polynomial fuction which is continuous when `x gt 0`.
At x= 0
L.H.L `=underset(xrarr0^-)lim f(x)=underset(hrarr0)limf(0-h)`
`=underset(hrarr0)lim(sin (-h))/((-h))=1`
R. H.L `=underset(xrarr0^+)lim f(x)==underset(hrarr0)limf(0+h)`
`=underset(hrarr0)lim(h+2)=2`
`therefore L.H.L ne R.H.L.`
`rArr ` f(x) is not continuous at x=0.
Therefore f(x) is continuous for all real values of x excepts '0'.