Chapter 2: Q20E (page 77)
Find the limit or show that it does not exist.
20. \(\mathop {\lim }\limits_{x \to \infty } \frac{{3{x^3} - 8x + 2}}{{4{x^3} - 5{x^2} - 2}}\)
The value of the limit is \(\frac{3}{4}\).
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\(y = c{x^{\rm{2}}}\), \({x^{\rm{2}}} + {\rm{2}}{y^{\rm{2}}} = k\).
\({x^{\rm{2}}} + {y^{\rm{2}}} = ax\), \({x^{\rm{2}}} + {y^{\rm{2}}} = by\).
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?
45. \(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}}{x + {\bf{2}}}&{{\bf{if}}\,\,\,x < {\bf{0}}}\\{{e^x}}&{{\bf{if}}\,\,\,{\bf{0}} \le x \le {\bf{1}}}\\{{\bf{2}} - x}&{{\bf{if}}\,\,\,x > {\bf{1}}}\end{array}} \right.\)
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 \(\mathop {{\bf{lim}}}\limits_{x \to - \infty } g\left( x \right) = - \infty \).
Sketch the graph of the function fwhere the domain is \(\left( { - {\bf{2}},{\bf{2}}} \right)\),\(f'\left( {\bf{0}} \right) = - {\bf{2}}\), \(\mathop {{\bf{lim}}}\limits_{x \to {{\bf{2}}^ - }} f\left( x \right) = \infty \), f is continuous at all numbers in its domain except \( \pm {\bf{1}}\), and f is odd.
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