Find the value of a constant V so that the function k(x) is continuous for all x k(x)= ) x+v&xlt 0 4-x^2&xgeqslant 0

To ensure that the function $k(x)$ is continuous for all $x$, the left-hand limit and the right-hand limit at $x=0$ must be equal. <br /><br />First, let's find the left-hand limit as $x$ approaches 0:<br />$\lim_{{x \to 0^-}} k(x) = \lim_{{x \to 0^-}} (x+v) = v$<br /><br />Next, let's find the right-hand limit as $x$ approaches 0:<br />$\lim_{{x \to 0^+}} k(x) = \lim_{{x \to 0^+}} (4-x^2) = 4$<br /><br />For $k(x)$ to be continuous at $x=0$, the left-hand limit and the right-hand limit must be equal. Therefore, we set them equal to each other and solve for $v$:<br />$v = 4$<br /><br />So, the value of the constant $v$ that makes the function $k(x)$ continuous for all $x$ is $v = 4$.