explain why the function is discontinuous at the given number a. (select all that apply.)\n\n$$ f ( x ) =…

explain why the function is discontinuous at the given number a. (select all that apply.)\n\n$$ f ( x ) = left{ \begin{array} { l l } { \frac { x ^ { 2 } - 2 x } { x ^ { 2 } - 4 } } & { \text { if } x \neq 2 } \\ { 1 } & { \text { if } x = 2 } end{array} \right. $$\n\n$$ a = 2 $$\n\n- ( f ( 2 ) ) is undefined.\n- ( f ( 2 ) ) is defined and ( lim _ { x \rightarrow 2 } f ( x ) ) is finite, but they are not equal.\n- ( lim _ { x \rightarrow 2 } f ( x ) ) does not exist.\n- ( lim _ { x \rightarrow 2 ^ { + } } f ( x ) ) and ( lim _ { x \rightarrow 2 ^ { - } } f ( x ) ) are finite, but are not equal.\n- none of the above\n\nsketch the graph of the function.

explain why the function is discontinuous at the given number a. (select all that apply.)\n\n$$ f ( x ) = left{ \begin{array} { l l } { \frac { x ^ { 2 } - 2 x } { x ^ { 2 } - 4 } } & { \text { if } x \neq 2 } \\ { 1 } & { \text { if } x = 2 } end{array} \right. $$\n\n$$ a = 2 $$\n\n- ( f ( 2 ) ) is undefined.\n- ( f ( 2 ) ) is defined and ( lim _ { x \rightarrow 2 } f ( x ) ) is finite, but they are not equal.\n- ( lim _ { x \rightarrow 2 } f ( x ) ) does not exist.\n- ( lim _ { x \rightarrow 2 ^ { + } } f ( x ) ) and ( lim _ { x \rightarrow 2 ^ { - } } f ( x ) ) are finite, but are not equal.\n- none of the above\n\nsketch the graph of the function.

Answer

Explanation:

Step1: Calculate (f(2))

Given (f(x)=\begin{cases}\frac{x^{2}-2x}{x^{2}-4},&x\neq2\1,&x = 2\end{cases}), so (f(2)=1).

Step2: Simplify (\frac{x^{2}-2x}{x^{2}-4})

Factor numerator and denominator: (\frac{x(x - 2)}{(x + 2)(x - 2)}=\frac{x}{x + 2}(x\neq2)).

Step3: Calculate (\lim_{x\rightarrow2}f(x))

(\lim_{x\rightarrow2}f(x)=\lim_{x\rightarrow2}\frac{x}{x + 2}=\frac{2}{2+2}=\frac{1}{2}).

Since (f(2) = 1) and (\lim_{x\rightarrow2}f(x)=\frac{1}{2}), (f(2)) is defined and (\lim_{x\rightarrow2}f(x)) is finite, but they are not equal.

Answer:

(f(2)) is defined and (\lim_{x\rightarrow2}f(x)) is finite, but they are not equal.