Question

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

Answer 1

$3-kx$ and $\frac{x^2}{4}-3$ are continuous all over their domains because these are polynomials, thus $F$ is continuous all over $[1;2[\cup]2;\infty[$ for an arbitrary $k$.

So we need to find $\lim\limits_{x\to 2-0}3-kx$ and $\lim\limits_{x\to 2+0}\left(\frac{x^2}{4}-3\right)$ and set them equal to achieve $F$ to be continuous at $x=2$ . So

$\lim\limits_{x\to 2-0}3-kx=3-2k$

$\lim\limits_{x\to 2+0}\left(\frac{x^2}{4}-3\right)=\frac{2^2}{4}-3=1-3=-2$ , so

$3-2k=-2$

$k=\frac52$