(Solved): Problem 4: Limits and Continuity for a Piecewise Function (18 points) For this problem, we will us ...

Problem 4: Limits and Continuity for a Piecewise Function (18 points) For this problem, we will use all of our techniques to explore the continuity properties of the piecewise function $$h(x)=?\begin{array}{ll}\frac{x^2+2x?8}{x?2}& \text{ if }x<2\\ 5& \text{ if }x=2\\ 3\sqrt{x+2}& \text{ if }x>2\end{array}$$ First we need to evaluate the limit as $$x$$ approaches 2 . a) (4 points) Evaluate $$\underset{x?2^{+}}{\lim }h(x)$$. EXPLAIN citing any limit laws used, similar to Problem 2(a). $$\underset{x?2^{+}}{\lim }h(x)=$$ b) (6 points) Evaluate $$\underset{x?2^{?}}{\lim }h(x)$$. In this part individual limit laws do not need to be cited, but you must check the form and show your work, similar to Problem 3(b). Form: $$\underset{x?2^{?}}{\lim }h(x)=$$ c) (4 points) Evaluate $$\underset{x?2}{\lim }h(x)$$. EXPLAIN using your answers to Problems 4(a) and 4(b) and the definition of a two-sided limit. $$\underset{x?2}{\lim }h(x)=$$ Now that we have evaluated the two-sided limit, we can check continuity at $$x=2$$. d) (4 points) Is $$h(x)$$ continuous at $$x=2$$ ? EXPLAIN using the definition of continuity.