Chapter 2: Q23E (page 77)
19-32 Prove the statement using the \(\varepsilon \), \(\delta \)definition of a limit.
23. \(\mathop {{\bf{lim}}}\limits_{x \to a} x = a\)
The given limit is true.
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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. \(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}}{{{\bf{2}}^x}}&{{\bf{if}}\,\,\,x \le {\bf{1}}}\\{{\bf{3}} - x}&{{\bf{if}}\,\,\,{\bf{1}} < x \le {\bf{4}}}\\{\sqrt x }&{{\bf{if}}\,\,\,x > {\bf{4}}}\end{array}} \right.\)
A particle moves along a straight line with the equation of motion\(s = f\left( t \right)\), where s is measured in meters and t in seconds.Find the velocity and speed when\(t = {\bf{4}}\).
\(f\left( t \right) = {\bf{10}} + \frac{{{\bf{45}}}}{{t + {\bf{1}}}}\)
\(y = a{x^{\rm{3}}}\), \({x^{\rm{2}}} + {\rm{3}}{y^{\rm{2}}} = b\).
Explain the meaning of each of the following.
(a)\(\mathop {{\bf{lim}}}\limits_{x \to - {\bf{3}}} f\left( x \right) = \infty \)
(b)\(\mathop {{\bf{lim}}}\limits_{x \to {{\bf{4}}^ + }} f\left( x \right) = - \infty \)
19-32: Prove the statement using the \(\varepsilon ,\delta \) definition of a limit.
28. \(\mathop {\lim }\limits_{x \to - {6^ + }} \sqrt(8){{6 + x}} = 0\)
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