Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

34_Introduction to Analysis_[Ch. 1
The definition of an infinite f (x) as x > co is analogous. As in the case
of infinitesimals, we introduce the concept of infinites of different orders.
288. Prove that the function
is an infinitesimal as x *oo. For what values of x is the ine-
quality
l/WI1. For what values of x is the ine-
quality
!/(*)!<
fulfilled if e is an arbitrary positive number? Calculate numerically
for: a) e-0.1; b) e = 0.01; c) e = 0.001.
290. Prove that the function
~
x 2
is an infinite for x *2. In what neighbourhoods of |x 2|N
fulfilled if N is an arbitrary positive number?
Find 5 if a) #=10; b) #=100;
J2^ c) #=1000.
o 291. Determine the order of smallness
of: a) the surface of a sphere, b) the volume
of a sphere if the radius of the sphere r
is an infinitesimal of order one. What
will the orders be of the radius of the
sphere and the volume of the sphere with
respect to its surface?
292. Let the central angle a of a circular
sector ABO (Fig. 9) with radius R
tend to zero. Determine the orders of
the infinitesimals relative to the
chord AB\ b)
infinitesimal a: a) of the
of the line CD; c) of the area of A/4BD.