Chapter 2: Q16E (page 77)
Find the limit or show that it does not exist.
16. \(\mathop {\lim }\limits_{x \to \infty } \frac{{ - 2}}{{3x + 7}}\)
The value of the limit is 0.
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(a). Prove Theorem 4, part 3.
(b). Prove Theorem 4, part 5.
The point \(P\left( {{\bf{1}},{\bf{0}}} \right)\) lies on the curve \(y = {\bf{sin}}\left( {\frac{{{\bf{10}}\pi }}{x}} \right)\).
a. If Qis the point \(\left( {x,{\bf{sin}}\left( {\frac{{{\bf{10}}\pi }}{x}} \right)} \right)\), find the slope of the secant line PQ (correct to four decimal places) for \(x = {\bf{2}}\), 1.5, 1.4, 1.3, 1.2, 1.1, 0.5, 0.6, 0.7, 0.8, and 0.9. Do the slopes appear to be approaching a limit?
b. Use a graph of the curve to explain why the slopes of the secant lines in part (a) are not close to the slope of the tangent line at P.
c. By choosing appropriate secant lines, estimate the slope of the tangent line at P.
(a) The curve with the equation \({y^2} = 5{x^4} - {x^2}\)is called akampyle of Eudoxus. Find and equation of the tangent line to this curve at the point\(\left( {1,2} \right)\)
(b) Illustrate part\(\left( a \right)\)by graphing the curve and the tangent line on a common screen. (If your graph device will graph implicitly defined curves, then use that capability. If not, you can still graph this curve by graphing its upper and lower halves separately.)
Show, using implicit differentiation, that any tangent line at a point\(P\) to a circle with center\(c.\) is perpendicular to the radius \(OP.\)
Calculate each of the limits
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