Find the values of a and b that make f continuous everywhere.

Question

f(x) = (x² − 4)/(x − 2) if x < 2

ax² − bx + 3 if 2 ≤ x< 3

4x − a + b if x ≥ 3

Answer 1

take the first 2 equations and set them equal to each other. Also, plugin x value and solve for missing variable

now take the 2nd and 3rd equations and do the same.....this way equations will be continuous...

you might get a system of equations then solve using substitution and elimination.

Answer 2

You impose the right and left limits as x approaches 2 and 3 to be equal.

Hence,

lim as x-->2^- of f(x)=lim as x-->2^- (x+2) = 4 (after writing (x²-4)/(x-2) = (x-2)(x+2)/(x-2)=x+2

and this must be equal to

lim as x-->2⁺ of f(x)=lim as x-->2⁺ ax²-bx +3 =4a-2b+3

The continuity at x=2 is then implied by the condition 4a-2b+3=4 which is 4a-2b=1.

In the same way you must impose

lim as x-->3^- of f(x)=lim as x-->3^- ax²-bx +3 = 9a-3b+3

and this must be equal to

lim as x-->3⁺ of f(x)=lim as x-->3⁺ 4x-a+b =12-a+b

The continuity at x=2 is then implied by the condition 9a-3b+3=12-a+b which is 10a-4b=9.

Therefore, you have the following system of equations for a and b

4a-2b=1 and 10a-4b=9 which can be solved to give you a=7/2 and b=13/2

Best,

Davide