For what value of the constant c is the function f continuous on (negative infin

Question

For what value of the constant c is the function f continuous on (negative infinity, infinity)?

for f(x) = cx^2+2x if x<2
x^3-cx if x> or = 2

Can someone pleeeaaaase help me????

Explanation / Answer

as ... for a function to be continuous on an interval, 3 things must be defined 1. lim (x -> a) f(x) must be defined 2. f(a) must be defined 3. lim(x -> a) f(x) = f(a) To find what value of c makes this function continuous on this interval, we must find these 3 things. Recall also that lim (x -> a) is defined if both the left-hand and right-hand limits are the same for that x value. First, we must consider the limit from both the left and right of x = 2. We'll start by taking the limit as x approaches 2 from the left. lim (x -> 2-) f(x) = lim (x -> 2-) [cx^2 + 2x] = 4c + 4 Now the limit as x approaches 2 from the right: lim (x -> 2+) f(x) = lim (x -> 2+) [x^3 - cx] = 8 - 2c It can also be seen that f(2) is defined as f(2) = 8 - 2c. Since the function is continuous everywhere on the real interval, we now set the limits equal to each other and solve for c. 4c + 4 = 8 - 2c 6c = 4 c = 2/3 So, with c = 2/3, this function is defined everywhere on the real domain.

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