(1) since `lim_(x->oo) x-1/2+x^2 = oo`

and `sin x` does not have a limit as `x->oo` because it does not approach any number but varies between -1 and 1, then

`lim_(x->oo) sin(x-1/2+x^2)` does not exist

(2) `lim_(x->-1^+) sqrt(x+1)-1=lim_(x->-1^+) sqrt(x+1) - lim_(x->-1^+)1`

from the right side `lim_(x->-1+) sqrt(x+1) = 0` ,...

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(1) since `lim_(x->oo) x-1/2+x^2 = oo`

and `sin x` does not have a limit as `x->oo` because it does not approach any number but varies between -1 and 1, then

`lim_(x->oo) sin(x-1/2+x^2)` does not exist

(2) `lim_(x->-1^+) sqrt(x+1)-1=lim_(x->-1^+) sqrt(x+1) - lim_(x->-1^+)1`

from the right side `lim_(x->-1+) sqrt(x+1) = 0` , and `lim_(x->-1^+) = 1`

So `lim_(x->-1^+) sqrt(x+1)-1 = 0 - 1 = -1`

(3) Really the same issue as (1) `lim_(x->oo) x^2+1/x+1 = oo` so the limit of sin does not exist for the same reason as in (1).