Which of the following functions is continuous at x = 2? Select one: a. f of x equals the quotient of the quantity x squared minus 4 and the quantity x minus 2 b. f of x equals the quotient of the quantity x squared minus 4 and the quantity x minus 2 for x not equal to 2 and equals 3 for x equals 2 c. f of x equals the quotient of the quantity x squared minus 4 and the quantity x minus 2 for x not equal to 2 and equals 4 for x equals 2 d. All are continuous at x = 2

Answer:option (a) and (c).Step-by-step explanation:(a) The function f(x) is given as:[tex]f(x)=\dfrac{x^2-4}{x-2}[/tex]which could also be represented as:[tex]f(x)=\dfrac{(x-2)(x+2)}{x-2}\\\\f(x)=x+2[/tex]clearly the limit of the function at x=2 exist  (since f(x)=x+2, and limit x→2 f(x)=2+2=4)and also f(x) is a polynomial function and hence is continuous at x=2.(b) The function f(x) is given as:[tex]f(x)=\dfrac{x^2-4}{x-2}=x+2[/tex] when x≠2 (  done in part (a))[tex]f(x)=3[/tex] when x=2for checking the continuity of a function at a point x=a we must know that:The left hand limit(L.H.L) of f(x)(at x=a)=Right hand limit(R.H.L.) of f(x)(at x=a)=f(a)so from here we get that L.H.L=R.H.L=4 (since f(x)=x+2,and limit x→2 f(x)=2+2=4)but f(2)=3Hence, the function is not continuous.(c) The function f(x) is given as:[tex]f(x)=\dfrac{x^2-4}{x-2}=x+2[/tex] when x≠2[tex]f(x)=4[/tex] when x=2Now the L.H.L=R.H.L=4 (since f(x)=x+2, and limit x→2 f(x)=2+2=4)Also f(2)=4Hence as L.H.L=R.H.L=f(2)Hence, the function is continuous.(d) As function in part (b) is not continuous.Hence option (d) is incorrect ( All are continuous at x=2)Hence, the answer to this question are option (a) and (c).