Note that f is continuous on (−∞, 6) and (6, ∞). For the function to be continuous on (−∞, ∞), we need to ensure that as x approaches 6, the left and right limits match.
First we find the left limit.
lim
x→6−
f(x) =
lim
x→6−
(cx2 + 3x)
To find the left limit as x approaches 6, we replace x with a value slightly less than 6 in the expression for f(x), which is c*x^2 + 3*x. So, let's substitute x = 6 - h, where h is a small positive number close to 0.
lim(x→6-) f(x) = lim(x→6-) [c*(6 - h)^2 + 3*(6 - h)]
Now, let's simplify the expression:
lim(x→6-) f(x) = lim(x→6-) [c*(36 - 12h + h^2) + 18 - 3h]
= c*(36 - 12h + h^2) + 18 - 3h
Next, we take the limit as h approaches 0:
lim(x→6-) f(x) = c*(36 - 12*0 + 0^2) + 18 - 3*0
= c*36 + 18
So, the left limit of f(x) as x approaches 6 is c*36 + 18.
Now, let's find the right limit as x approaches 6.
lim(x→6+) f(x) =
lim(x→6+)
(cx^2 + 3x)
Similarly, we replace x with a value slightly greater than 6, so let's substitute x = 6 + h, where h is a small positive number close to 0.
lim(x→6+) f(x) = lim(x→6+) [c*(6 + h)^2 + 3*(6 + h)]
Again, we simplify the expression:
lim(x→6+) f(x) = lim(x→6+) [c*(36 + 12h + h^2) + 18 + 3h]
= c*(36 + 12h + h^2) + 18 + 3h
Taking the limit as h approaches 0:
lim(x→6+) f(x) = c*(36 + 12*0 + 0^2) + 18 + 3*0
= c*36 + 18
So, the right limit of f(x) as x approaches 6 is also c*36 + 18.
For the function to be continuous at x = 6, the left and right limits must be equal. Therefore, we need to ensure that:
c*36 + 18 = c*36 + 18
Simplifying the equation, we can see that the equation holds true for any value of c. Therefore, to ensure continuity of f(x) on (-∞, ∞), any value of c will work.