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Answer 1

$4) underset{nrightarrow infty}{lim} dfrac{(sqrt{n} + 1)^{2} - n}{sqrt{n}} = left(dfrac{infty}{infty} right) = underset{nrightarrow infty}{lim} dfrac{n + 2sqrt{n} + 1 - n}{sqrt{n}} = underset{nrightarrow infty}{lim} dfrac{2sqrt{n} + 1}{sqrt{n}} =\ = underset{nrightarrow infty}{lim} left(dfrac{2sqrt{n}}{sqrt{n}} + dfrac{1}{sqrt{n}} right) = underset{nrightarrow infty}{lim} left(2 + dfrac{1}{sqrt{n}} right) = 2 + dfrac{1}{infty} = 2 + 0 = 2$