Chapter 2: Q87E (page 77)
Evaluate
\(\mathop {{\bf{lim}}}\limits_{x \to {\bf{1}}} \frac{{{x^{{\bf{1000}}}} - {\bf{1}}}}{{x - {\bf{1}}}}\)
The value of the expression is 1000.
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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 \(\left( {{\rm{12,3}}} \right)\)
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.
Use thegiven graph of f to state the value of each quantity, if it exists. If it does not exists, explain why.
(a)\(\mathop {{\bf{lim}}}\limits_{x \to {{\bf{2}}^ - }} f\left( x \right)\)
(b)\(\mathop {{\bf{lim}}}\limits_{x \to {{\bf{2}}^ + }} f\left( x \right)\)
(c) \(\mathop {{\bf{lim}}}\limits_{x \to {\bf{2}}} f\left( x \right)\)
(d) \(f\left( {\bf{2}} \right)\)
(e) \(\mathop {{\bf{lim}}}\limits_{x \to {\bf{4}}} f\left( x \right)\)
(f) \(f\left( {\bf{4}} \right)\)
The gravitational force exerted by the planet Earth on a unit mass at a distance r from the center of the planet is
\(F\left( r \right) = \left\{ {\begin{array}{*{20}{c}}{\frac{{GMr}}{{{R^{\bf{3}}}}}}&{{\bf{if}}\,\,\,r < R}\\{\frac{{GM}}{{{r^{\bf{2}}}}}}&{{\bf{if}}\,\,\,r \ge R}\end{array}} \right.\)
Where M is the mass of Earth, R is its radius, and G is the gravitational constant. Is F a continuous function of r?
Which of the following functions \(f\) has a removable discontinuity at \(a\)? If the discontinuity is removable, find a function \(g\) that agrees with \(f\) for \(x \ne a\)and is continuous at \(a\).
(a) \(f\left( x \right) = \frac{{{x^4} - 1}}{{x - 1}},a = 1\)
(b) \(f\left( x \right) = \frac{{{x^3} - {x^2} - 2x}}{{x - 2}},a = 2\)
(c)\(f\left( x \right) = \left [{\sin x} \right],a = \pi \)
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