Calculus: Analytic Geometry and Calculus, with Vectors

(^126) Functions, limits, derivatives that e is small, we need not deny ourselves the satisfaction of the feeling that, when the ...

3.2 Limits 127 such that (3.272) lim (1 + x) If- = e. X-O Anybody can collect a little evidence in support of this assertion by ...

128 Functions, limits, derivatives Theorem 3.287 (sandwich theorem or flyswatter theorem) If for some positive number p g (x) &l ...

3.2 Limits 129 Problems 3.29 1 It is not enough to be able to read the four assertions which involve f(x) when x is near a but x ...

130 Functions, limits, derivatives that scientists of the future will adopt notation like that in (1) and that their historians ...

3.2 Limits 131 9 Supposing that y = x2 and y +i y = (x +Ax)2, show that 10 lim Ly = 2x. ,iz-.0 Ox Prove that lim (x + x3= 3x2. 1 ...

132 Functions, limits, derivatives 16 Recall that the signum function having values sgn x (read signum x) is defined by the form ...

3.3 Unilateral limits and asymptotes 133 3.3 Unilateral limits and asymptotes the function f for which f(x) = sgn (x - a) and se ...

134 Functions, limits, derivatives psychologically satisfying. Whenever a positive number a is selected, we can find a positive ...

3.3 Unilateral limits and asymptotes 135 that is, f(x) exceeds M whenever a < x < a + 6. The assertion that f(x) is large ...

136 Functions, limits, derivatives In this case, the line having the equation y = 4x + B is called an asymptote of the graph. Th ...

3.3 Unilateral limits and asymptotes 137 cosx lim L1 x2 x4-2i+¢i-6i+x3 +(_1)n(2n)!1x2n (v) n-. w .11 r x3 xs x7 n x2nt1 (a) sinx ...

(^138) Functions, limits, derivatives in which a and b are positive constants, is a hyperbola. (x, y(x)) lies on the hyperbola a ...

3.3 Unilateral limits and asymptotes 139 which is applicable when n is a positive integer, show that 1! = 1, 2! = 2, 3! = 6, 4! ...

140 Functions, limits, derivatives 15 Try to make friends of the contents of the preceding problems by proving that (1) ()+( kn ...

3.3 Unilateral limits and asymptotes 141 17 We can feel sure that if IxI < 1, then x" is near 0 whenever n is large. How can ...

142 Functions, limits, derivatives 23 Sometime we will learn that (1) lim 0. ,.2- Hence there must be an integer N such that (2) ...

3.3 Unilateral limits and aysmptotes 143 27 For hundreds of years, people have been interested in the magnitude of 7r(x), the nu ...

144 Functions, limits, derivatives relations 0 oo = 1, a = oo, and a = - oo are as absurd in the "algebra" of S* as the symbol a ...

3.4 Continuity 145 tion of limit. With a small change in notation, we can see that f is continuous at x if and only if (3.421) o ...