Chapter 2: Q15E (page 77)
15-42Find the limit or show that it does not exist.
15. \(\mathop {{\bf{lim}}}\limits_{x \to \infty } \,\,\frac{{{\bf{4}}x + {\bf{3}}}}{{{\bf{5}}x - {\bf{1}}}}\)
The value of the given expression is \(\frac{4}{5}\).
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19-32: Prove the statement using the \(\varepsilon ,\delta \) definition of a limit.
29. \(\mathop {\lim }\limits_{x \to 2} \left( {{x^2} - 4x + 5} \right) = 1\)
A roast turkey is taken from an oven when its temperature has reached \({\bf{185}}\;^\circ {\bf{F}}\) and is placed on a table in a room where the temperature \({\bf{75}}\;^\circ {\bf{F}}\). The graph shows how the temperature of the turkey decreases and eventually approaches room temperature. By measuring the slope of the tangent, estimate the rate of change of the temperature after an hour.
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\(t = {\bf{4}}\).
\(f\left( t \right) = {\bf{10}} + \frac{{{\bf{45}}}}{{t + {\bf{1}}}}\)
38: If H is the Heaviside function defined in section 2.2, prove, using Definition 2, that \(\mathop {\lim }\limits_{t \to 0} H\left( t \right)\) does not exist.
(Hint: Use an indirect proof as follows. Suppose that the limit is L. Take \(\varepsilon = \frac{1}{2}\) in the definition of a limit and try to arrive at a contradiction.)
19-32 Prove the statement using the \(\varepsilon \), \(\delta \)definition of a limit.
25. \(\mathop {{\bf{lim}}}\limits_{x \to 0} {x^{\bf{2}}} = {\bf{0}}\)
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