Evaluate `lim_(x->oo) sin((x^2+1)/(x+1))` and `lim_(x->pi+) cosx/(x-pi)`

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The limits` lim_(x->oo) sin((x^2+1)/(x+1))` and` lim_(x->pi+) cosx/(x-pi)` have to be determined.

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

As x tends to infinite `(x^2 + 1)/(x + 1)` tends to infinite as the numerator grows larger than the denominator at a faster rate.

As a result, the limit tends to ` sin(oo)` which is not defined but the value lies in [-1, 1]

`lim_(x->pi+) cosx/(x-pi)`

substituting x = pi gives `1/0 = oo` .

The limit ` lim_(x->oo) sin((x^2+1)/(x+1))` is undefined but bounded and `lim_(x->pi+) cosx/(x-pi) = oo`

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