Chapter 8 Sequences and Series of
f(x) = 0. To prove this, fix x, call it x0 E [0,1], and show that lim 1, (x0) = 0. Let s > 0 be given.
We need to nd n* E N such that |f(x0)fn(xo)| < 5
Chapter 9 Vector Calculus
2. Ifthe points are A(3,3,3), B(2,1,1), C(5,4, 2), then d(AB)=1/E,'d(B,C)= 3%, d(A,C)=1/g. Since,
[auto]2 +[d(A,B)]2 =[d(B, 6)]r2 , this is a right triangle.
4. Let (x, y,z) be a point equidistant from the given point
Chapter 2 Sequences
1 _ . . .
1. < 0.02 <21 < (0.02)1In +1 <2 n > 2499. Thus, 1f 11* = 2,500 , then for any n 2 11* the given inequahty
will be true.
2. (a) We prove that lim an = 0. Let s > 0 be given. We need to nd 11* EN 50 that Ian
Chapter 10 Functions of Two Variables
1. (a) Domain is the set of all points (Ly) such that :c2 + (y +1)2 5 1.
(c) Domain is the set of all points (x, y) in mg.
2. (a) We will prove that the value of the limit is 0. Let s > 0 be given and cho
Chapter 4 Continuity
2. (a) No, f is not continuous at x = 0 , thus not continuous at every point in the interval.
(b) Yes, f is right continuous at every point in the interval.
(c) Yes, f is continuous at every point in the interval.
Chapter 3 Limits of Functions
1. lim f (x ) = 00 if and only if for any real K > 0 there exists a real number M > 0 such that f (x) < K
provided x 2 M and x is in the domain of f.
2. (a) We will prove that lim f (x)= 2. Therefore, let a >