plain why the function is discontinuous at the given number a. (Select all that apply.) f(x) = x2 − 2x x2 − 4 if x ≠ 2 1 if x = 2 a = 2

Answer with Step-by-step explanation:We are given that[tex]f(x)=\left\{\begin{matrix}\dfrac{x^2-2x}{x^2-4}&,if\ \ x\neq 2 \\ 1&,if\ \ x=2\end{matrix}\right.[/tex]We have to explain that why the function is discontinuous at x=2We know that if function is continuous at x=a then LHL=RHL=f(a).[tex]f(x)=\frac{x(x-2)}{(x+2)(x-2)}=\frac{x}{x+2}[/tex]LHL=Left hand limit when x <2Substitute x=2-h where h is small positive value >0[tex]\lim_{h\rightarrow 0}f(x)=\lim_{h\rightarrow 0}\frac{2-h}{2-h+2}[/tex][tex]\lim_{h\rightarrow 0}\frac{2-h}{4-h}=\frac{2}{4}=\frac{1}{2}[/tex]Right hand limit =RHL when x> 2Substitute x=2+h[tex]\lim_{h\rightarrow 0}f(x)=\lim_{h\rightarrow 0}\frac{2+h}{2+h+2}=\lim_{h\rightarrow 0}\frac{2+h}{4+h}[/tex][tex]=\frac{2}{4}=\frac{1}{2}[/tex]LHL=RHL=[tex]\frac{1}{2}[/tex]f(2)=1[tex]LHL=RHL\neq f(2)[/tex]Hence, function is discontinuous at x=2