Chapter 2: Q15E (page 77)
Evalauate the limit, if it exists.
\(\mathop {{\bf{lim}}}\limits_{x \to {\bf{2}}} \frac{{{x^{\bf{2}}} + {\bf{5}}x + {\bf{4}}}}{{x - {\bf{2}}}}\)
The limit does not exist.
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43-45 Find the numbers at which f is discontinuous. At which of these numbers is f continuous from the right, from the left, or neither? Sketch the graph of f?
44. \(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}}{{{\bf{2}}^x}}&{{\bf{if}}\,\,\,x \le {\bf{1}}}\\{{\bf{3}} - x}&{{\bf{if}}\,\,\,{\bf{1}} < x \le {\bf{4}}}\\{\sqrt x }&{{\bf{if}}\,\,\,x > {\bf{4}}}\end{array}} \right.\)
Sketch the graph of the function gthat is continuous on its domain \(\left( { - {\bf{5}},{\bf{5}}} \right)\) and where\(g\left( {\bf{0}} \right) = {\bf{1}}\), \(g'\left( {\bf{0}} \right) = {\bf{1}}\), \(g'\left( { - {\bf{2}}} \right) = {\bf{0}}\), \(\mathop {{\bf{lim}}}\limits_{x \to - {{\bf{5}}^ + }} g\left( x \right) = \infty \), and \(\mathop {{\bf{lim}}}\limits_{x \to - {{\bf{5}}^ - }} g\left( x \right) = {\bf{3}}\).
Sketch the graph of the function g for which \(g\left( {\bf{0}} \right) = g\left( {\bf{2}} \right) = g\left( {\bf{4}} \right) = {\bf{0}}\), \(g'\left( {\bf{1}} \right) = g'\left( {\bf{3}} \right) = {\bf{0}}\), \(g'\left( {\bf{0}} \right) = g'\left( {\bf{4}} \right) = {\bf{1}}\), \(g'\left( {\bf{2}} \right) = - {\bf{1}}\), \(\mathop {{\bf{lim}}}\limits_{x \to \infty } g\left( x \right) = \infty \), and \(\mathop {{\bf{lim}}}\limits_{x \to - \infty } g\left( x \right) = - \infty \).
Each limit represents the derivative of some function f at some number a. State such as an f and a in each case.
\(\mathop {{\bf{lim}}}\limits_{h \to {\bf{0}}} \frac{{{e^{ - {\bf{2}} + h}} - {e^{ - {\bf{2}}}}}}{h}\)
Let\(f\left( x \right) = 1/x\), and \(g\left( x \right) = 1/{x^2}\).
(a) Find \(\left( {f \circ g} \right)\left( x \right)\).
(b) Is\(f \circ g\) continuous everywhere? Explain.
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