2.1.28 Calculation of limits

Problem: Find the limits of the function without using L'Hopital's rule. a) \( \lim _{x \rightarrow 1} \frac{2-x-x^{2}}{1-x} \), b) \( \lim _{x \rightarrow \infty} \frac{x^{3}+x^{2}+x+1}{2 x^{3}+4} \), c) \( \lim _{x \rightarrow 4} \frac{\sqrt{x}-2}{x-4} \), d) \( \lim _{x \rightarrow 0} \frac{\tan x-\sin x}{x^{3}} \), e) \( \lim _{x \rightarrow 0}(1+2 x)^{\frac{3}{x}} \) f) \( \lim _{x \rightarrow 0} \frac{\ln x}{\cot x} \).