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

option (a) and (c).

Step-by-step explanation:

(a) The function f(x) is given as:

which could also be represented as:

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:

when x≠2 (  done in part (a))

when x=2

for 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)=3

Hence, the function is not continuous.

(c) The function f(x) is given as:

when x≠2

when x=2

Now 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)=4

Hence 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).

problem:

f(x) = - 2/3x + 5; f(5/2)

substitute 5/2 in for x:

f(5/2) = -2/3(5/2) + 5

simplify the right side:

f(5/2) = -5/3 + 5

f(5/2) = 10/3

f(5/2) = 10/3

i hope this !