Question

Let f(x)= (x^2-2x)/(x^2-4)

For the function above, please find the following:

a) The domain of the function

b) The real zeroes of the function

c) The limit as × approaches 2 of f(x).

d) Plot f(x)

Answer 1

$$f(x)=\dfrac{x^2-2x}{x^2-4}$$

a)

$$x^2-4\not=0=>(x-2)(x+2)\not=0$$

$$x\not=-2, x\not=2$$

Domain: $(-\infin, -2)\cup (-2, 2)\cup (2, \infin)$

b)

$$f(x)=0=>\dfrac{x^2-2x}{x^2-4} =0$$

$$x(x-2)=0, x\not=-2, x\not=2$$

Zeroes: $x=0$

c)

$$\lim\limits_{x\to 2}f(x)=\lim\limits_{x\to 2}\dfrac{x^2-2x}{x^2-4}=\lim\limits_{x\to 2}\dfrac{x(x-2)}{(x-2)(x+2)}$$

$$=\lim\limits_{x\to 2}\dfrac{x}{x+2}=\dfrac{2}{2+2}=\dfrac{1}{2}$$

d)