Preview Activity 1.7.1.
A function \(f\) is given by the graph in the following figure. Use the graph to answer each of the following questions. Note: to the right of \(x = 2\text{,}\) the graph of \(f\) is exhibiting infinite oscillatory behavior similar to the function \(\sin(\frac{\pi}{x})\) that we encountered in the key example early in Section 1.2. Assume that \(f(2) = -2.5\text{.}\)
(a)
For each of the values \(a = -3, -2, -1, 0, 1, 2, 3\text{,}\) determine whether or not \(\lim_{x \to a} f(x)\) exists. If the function has a limit \(L\) at a given point, state the value of the limit using the notation \(\lim_{x \to a} f(x) = L\text{.}\) If the function does not have a limit at a given point, write a sentence to explain why.
(b)
For each of the values of \(a\) from part (a) where \(f\) has a limit, determine the value of \(f(a)\) at each such point. In addition, for each such \(a\) value, does \(f(a)\) have the same value as \(\lim_{x \to a} f(x)\text{?}\)
(c)
For each of the values \(a = -3, -2, -1, 0, 1, 2, 3\text{,}\) determine whether or not \(f'(a)\) exists. In particular, based on the given graph, ask yourself if it is reasonable to say that \(f\) has a tangent line at \((a,f(a))\) for each of the given \(a\)-values. If so, visually estimate the slope of the tangent line to find the value of \(f'(a)\text{.}\)
