2.1.27 Calculation of limits

Problem: Find the limits of the functions without using L'Hopital's rule. a) \( \lim _{x \rightarrow 3} \frac{x^{2}-9}{x^{2}+2 x-15} \), b) \( \lim _{x \rightarrow+\infty} \frac{\sqrt{x}+1}{\sqrt{x}} \), c) \( \lim _{x \rightarrow \infty}\left(\sqrt{x^{2}+1}-\sqrt{x^{2}-1}\right) \), d) \( \lim _{x \rightarrow 0} \frac{5 x^{2}}{\arcsin 3 x} \), e) \( \lim _{x \rightarrow \infty}\left(\frac{x-8}{x}\right)^{\frac{x}{2}} \), f) \( \lim _{x \rightarrow a} \frac{x^{m}-a^{m}}{x^{n}-a^{n}} \).