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`f(x) {{:(x + 1"," if x ge 1 ),(x^(2)+ 1"," if x lt 1):}` <br> If x `gt` 1 then f(x) + 1 is a polynomial function . <br> If x `lt` 1 then f(x) `x^(2)+ 1` is a polynomial funcation. <br> A polynomial function is always continous. <br> at x = 1 <br> f(1) = 1+ 1= 2 <br> L.H.L. ` underset (x to 1^(-))(lim)f(x)` <br> Let 1 - h = x <br> `rArr 1- h to 1` <br> `rArr h to 0` <br> `=underset (h to 0)(lim) f(1-h)` <br> `=underset (h to 0)(lim) (1 - h)^(2)+ 1` <br> ` (1 - 0) ^(2) + 1 = 1 + 1= 2` <br> R.H.L = ` underset (x to 1^(+))(lim)f(x)` <br> Let 1 - h = x <br> `rArr 1 - h to 1` <br> `rArr h to 2` <br> `=underset (h to 0)(lim) f(1+h)` <br> `=underset (h to 0)(lim) (1 - h)+ 1` <br> `(1 + 0) + 1 = 2 ` <br> `:'` L.H.L = f(1) = R.H.L <br> `:.` f(x) is continuous at x = 1 <br> Therefore, f(x) is always continuous . **Revision of limits**

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