For what value of the constant c is the function f continuous at x=2

Question

f(x) = {cx2+2x if x<2 , x^3 -cx if x>_2

Answer 1

For f(x) to be continuous at x = 2:

1. f(2) must exist
2. lim_x->2f(x) must exist
3. It must be true that f(2) = lim_x->2f(x)

f(2) will clearly exist no matter the value of c (polynomial function).

For the limit to exist the limit from the left of 2 must equal the limit from the right of 2.

To determine the limit from the left of 2 direct substitution into the rule for x < 2 is used.

lim_x->2- f(x) = c(2)² + 2(2) = 4c + 4

To determine the limit from the right of 2 use direct substitution into the rule for x≥ 2.

lim_x->2+ f(x) = (2)³ - 2c = 8 - 2c

Determine the value of c that makes limit from left equal limit from right:

4c + 4 = 8 - 2c

6c = 4

c = 2/3

When c = 2/3, f(2) = (2)³ - (2/3)(2) = 8 - 4/3 = 20/3

Also, when x < 2, f(x) = (2/3)x² + 2x

You can substitute 2 into this rule to determine the limit from the left of 2.

(2/3)(2)² + 4 = 8/3 + 12/3 = 20/3, i.e. limit from left equals limit from right, and limit as x approaches 2 is equal to f(2).

Note that this piecewise function is NOT differentiable at x = 2. The derivatives of the two pieces have different values at x = 2, so the slopes of tangent lines as x -> 2 have different values depending on approaching 2 from the left or right.