\( \begin{array}{l}\text { Sketch a graph of a tunction that contains } \\ \text { the following: } \\ \text { a) } f(0)=1 \\ \text { b) } \lim _{x \rightarrow 0^{-}} f(x)=-1 \\ \text { c) } \lim _{x \rightarrow 0^{+}} f(x)=2\end{array} \)
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To create a graph that meets the specified criteria, start by planning three key points around \( x = 0 \). Begin with \( f(0) = 1 \) which means at the point (0, 1), you will have a solid dot. As \( x \) approaches 0 from the left (\( 0^{-} \)), the function should drop down to -1, so you can depict this with a line sloping downwards that ends just before hitting \( (-1) \) on the y-axis as it approaches 0. For \( 0^{+} \), the function should rise sharply up to 2, and again, you can depict this with a rising line that begins just after \( 0 \). The graph should visually show the jump at \( x = 0 \); we'll call this a jump discontinuity where the left-hand limit is -1 and the right-hand limit is 2, with the function actually taking the value 1 at \( x = 0 \). This creates a clear visual representation of these limits and the actual function value at that point. Now, you have a graph with intriguing behavior around \( x = 0 \). It's a beautiful representation of how functions can behave unexpectedly, and it illustrates continuity and discontinuity nicely!
