a) $\lim_{x \to 1} \frac{x - \sqrt{x}}{\sqrt{x+8} - 3}$

Ta có dạng $\frac{0}{0}$ khi $x \to 1$. 

\begin{align*} \label{eq:1} \lim_{x \to 1} \frac{x - \sqrt{x}}{\sqrt{x+8} - 3} &= \lim_{x \to 1} \frac{\sqrt{x}(\sqrt{x} - 1)(\sqrt{x} + 1)(\sqrt{x+8} + 3)}{(\sqrt{x+8} - 3)(\sqrt{x+8} + 3)(\sqrt{x} + 1)} \\ &= \lim_{x \to 1} \frac{\sqrt{x}(x-1)(\sqrt{x+8} + 3)}{(x+8 - 9)(\sqrt{x} + 1)} \\ &= \lim_{x \to 1} \frac{\sqrt{x}(x-1)(\sqrt{x+8} + 3)}{(x-1)(\sqrt{x} + 1)} \\ &= \lim_{x \to 1} \frac{\sqrt{x}(\sqrt{x+8} + 3)}{\sqrt{x} + 1} \\ &= \frac{\sqrt{1}(\sqrt{1+8} + 3)}{\sqrt{1} + 1} \\ &= \frac{1(3+3)}{2} = 3\end{align*}

b) $\lim_{x \to -1} \frac{\sqrt{x+2} - 1}{\sqrt{x+5} - 2}$

Ta có dạng $\frac{0}{0}$ khi $x \to -1$. 

\begin{align*} \lim_{x \to -1} \frac{\sqrt{x+2} - 1}{\sqrt{x+5} - 2} &= \lim_{x \to -1} \frac{(\sqrt{x+2} - 1)(\sqrt{x+2} + 1)(\sqrt{x+5} + 2)}{(\sqrt{x+5} - 2)(\sqrt{x+5} + 2)(\sqrt{x+2} + 1)} \\ &= \lim_{x \to -1} \frac{(x+2 - 1)(\sqrt{x+5} + 2)}{(x+5 - 4)(\sqrt{x+2} + 1)} \\ &= \lim_{x \to -1} \frac{(x+1)(\sqrt{x+5} + 2)}{(x+1)(\sqrt{x+2} + 1)} \\ &= \lim_{x \to -1} \frac{\sqrt{x+5} + 2}{\sqrt{x+2} + 1} \\ &= \frac{\sqrt{-1+5} + 2}{\sqrt{-1+2} + 1} \\ &= \frac{2+2}{1+1} = 2 \end{align*}