Find the value of the constant aa that makes the following f

Find the value of the constant aa that makes the following function continuous on (,).

f(x)={(6x^317x^2+6x27)/(x3)   if x<3

{6x^2x+a   if x3

a=

Solution

for the function f(x) to be continuous at x = c

lim _{[x -> c^-]} f(x) = lim _{[x -> c⁺]} f(x) = f(c)

Given function is continuous in (,)

==> function is continuous at x = 3

==> lim _{[x -> 3^-]} f(x)

==> lim _{[x -> 3^-]} (6x³ - 17x² + 6x - 27)/(x - 3)   since for the values x < 3 , f(x) = (6x³ - 17x² + 6x - 27)/(x - 3)

we get 0/0 form hence apply L\'hospital rule . i.e, differentiate numerator and denominator with respect to x

==> lim _{[x -> 3^-]} (6(3)x^3-1 - 17(2)x^2-1 + 6(1) -0)/(1)

==> lim _{[x -> 3^-]} 18x² - 34x + 6

==> 18(3)² - 34(3) + 6

==> 66

==> lim _{[x -> 3⁺]} f(x)

==> lim _{[x -> 3⁺]} 6x² x + a    since for the value x > 3 , f(x) = 6x² x + a

==> 6(3)² - 3 + a

==> 51 + a

lim _{[x -> 3^-]} f(x) = lim _{[x -> 3⁺]} f(x)

==> 66 = 51 + a

==> a = 66 - 51

==> a = 15

Hence the value of must be 15 for the function to be continuous on (,)