Chapter 2: Q34E (page 77)
Determine the infinite limit.
\(\mathop {\lim }\limits_{x \to {0^ + }} \,\ln \left( {\sin x} \right)\)
The limit tends to negative infinity.
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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.)
Find \(f'\left( a \right)\).
\(f\left( t \right) = \frac{{\bf{1}}}{{{t^{\bf{2}}} + {\bf{1}}}}\)
41-42 Show that f is continuous on \(\left( { - \infty ,\infty } \right)\).
\(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}}{{\bf{1}} - {x^{\bf{2}}}}&{{\bf{if}}\,\,x \le {\bf{1}}}\\{{\bf{ln}}\,x}&{{\bf{if}}\,\,\,x > {\bf{1}}}\end{array}} \right.\)
The point \(P\left( {{\bf{0}}.{\bf{5}},{\bf{0}}} \right)\) lies on the curve \(y = {\bf{cos}}\pi x\).
(a) If Q is the point \(\left( {x,{\bf{cos}}\pi x} \right)\), find the slope of the secant line PQ (correct to six decimal places) for the following values of x:
(i) 0 (ii) 0.4 (iii) 0.49 (iv) 0.499 (v) 1 (vi) 0.6 (vii) 0.51 (viii) 0.501
(b) Using the results of part (a), guess the value of the slope of tangent line to the curve at \(P\left( {{\bf{0}}.{\bf{5}},{\bf{0}}} \right)\).
(c) Using the slope from part (b), find an equation of the tangent line to the curve at \(P\left( {{\bf{0}}.{\bf{5}},{\bf{0}}} \right)\).
(d) Sketch the curve, two of the secant lines, and the tangent line.
Find equations of both the tangent lines to the ellipse\({{\rm{x}}^{\rm{2}}}{\rm{ + 4}}{{\rm{y}}^{\rm{2}}}{\rm{ = 36}}\)that pass through the point \(\left( {{\rm{12,3}}} \right)\)
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