$\begin{array}{l}1a)\quad \mathop{\lim}\limits_{x \to 1}(x^2 - 3x + 1)= 1^2 - 3.1 + 1 =-1 \\ b)\quad \lim(\sqrt{n^2 + 2n} - n)\\ = \lim\left[\dfrac{(\sqrt{n^2 + 2n} - n)(\sqrt{n^2 + 2n} + n)}{\sqrt{n^2 + 2n} + n}\right]\\ = \lim\left(\dfrac{2n}{\sqrt{n^2 + 2n} + n} \right)\\ = \lim\left(\dfrac{2}{\sqrt{1 + \dfrac{2}{n}} + 1}\right)\\ = \dfrac{2}{\sqrt 1 + 1} = 1\\ 2a)\quad f(x) = \dfrac{2x - 2}{x-1}\\ \to f(x) = \dfrac{2(x-1)}{x-1}\\ \to f(x) = 2\\ \to f'(x) = 0\\ b)\quad f(x) = 2\sin x + \cos x\\ \to f'(x) = 2\cos x - \sin x \end{array}$
