Continuity of Piecewise Functions with constants

Question

I have a piecewise function:

$$f(x) = \begin{cases} x^2-3x, &x < 4\\ j, &x = 4\\ k+x, &x > 4 \end{cases}$$

I want to find the values of $j$ and $k$ that makes my function continuous where $x = 4$.

How do I go about finding $j$ and $k$?

Answer 1

$f(x)$ is continuous at $x = 4$ if and only if $$\lim_{x \to 4} f(x) = f(4) $$

In order for the limit to exist, we must have: \begin{align*} \lim_{x \to 4^-} f(x) &= \lim_{x \to 4^+} f(x) \\ \lim_{x \to 4^-} \left[x^2 - 3x \right] &= \lim_{x \to 4^+} \left[k + x \right] \\ 4^2 - 3(4) &= k + 4 \\ 4 &= k+4 \\ k &= 0 \end{align*}

So in order for $f(x)$ to be continuous, we have: $$ f(x) = \begin{cases} x^2-3x & x < 4\\ j & x = 4\\ x & x > 4 \end{cases} $$ and we see that: $$ \lim_{x \to 4} f(x) = 4 $$ Since continuity implies that the limit must equal $f(4) = j$, we have that $j = 4$.

Answer 2

J=4,K=0. Here you may use the LHS=RHS case.then you must find it out