Question

............................. ​

Answer 1

$y=\dfrac{ax+4}{2x+b}\ \ ,\ \ \ x=2\ ,\ y=1\ \ -\ \ asimptotu$

$y=1\ -\ asimptota\ \ \Rightarrow \ \ \lim\limits _{x \to \infty}\dfrac{ax+4}{2x+b}= \lim\limits _{x \to \infty}\dfrac{a+\frac{4}{x}}{2+\frac{b}{x}}=\dfrac{a}{2}=1\ \ \Rightarrow \\\\a=2\\\\ x=2\ -\ asimptota\ \ \Rightarrow \ \ \lim\limits _{x \to 2}\dfrac{2x+4}{2x+b}=\infty \ \ \Rightarrow \ \ \dfrac{2\cdot 2+4}{2\cdot 2+b}=\infty \ \ \Rightarrow \\\\4+b=0\ \ \ \Rightarrow \ \ \ b=-4$

$y=\dfrac{2x+4}{2x-4}=\dfrac{x+2}{x-2}=\dfrac{(x-2)+2+2}{x-2}=1+\dfrac{4}{x-2}\\\\\\n+\dfrac{k}{x+m}=1+\dfrac{4}{x-2}\ \ \ \Longrightarrow \ \ \ \ \underline {\ n=1\ ,\ k=4\ ,\ m=-2\ }\\\\\\\boxed {\ y=1+\dfrac{4}{x-2}\ }$