Question

c) lim\(\dfrac{\sqrt{x+3}-3}{x-6}\)(x-->6)

d) lim\(\dfrac{2x-6}{4-x}\)(x-->+\(\infty\))

Answer 1

\(\lim\limits_{x\rightarrow6}\dfrac{\sqrt{x+3}-3}{x-6}=\lim\limits_{x\rightarrow6}\dfrac{\left(\sqrt{x+3}-3\right)\left(\sqrt{x+3}+3\right)}{\left(x-6\right)\left(\sqrt{x+3}+3\right)}=\lim\limits_{x\rightarrow6}\dfrac{x-6}{\left(x-6\right)\left(\sqrt{x+3}+3\right)}\)

\(=\lim\limits_{x\rightarrow6}\dfrac{1}{\sqrt{x+3}+3}=\dfrac{1}{6}\)

\(\lim\limits_{x\rightarrow-\infty}\dfrac{2x-6}{4-x}=\lim\limits_{x\rightarrow-\infty}\dfrac{2-\dfrac{6}{x}}{\dfrac{4}{x}-1}=\dfrac{2}{-1}=-2\)