Answer to Question #92835 in Calculus for MGM

Question

For which values of k, is the function f, defined as below, continuous at x = 2 ?
F(x) = 3-kx, 1≤ x is < 2.
(X^2/4) -3 ,x ≥ 2
Further, at which other points in [1,∞[ is f continuous, and why?

Answer 1

Answer

For k=2.5.

Explanation

First of all, the defined function is continuous at every point of [1,2[ and ]2,$\infty$ [, because all polynomials are continuous at every point x $\isin \reals$ .

Then, the function f is defined at the point x=2, f(2)=2²/4-3=-2, and its right limit at this point is equal to -2 too. The left limit of the function at x= 2 also exists and it is equal to 3-k*2. For the continuity of the function, the left and right limits at the point 2 must be equal: 3-k*2 = -2. Hence k=2.5.