1549901369-Elements_of_Real_Analysis__Denlinger_

4.2 Algebra of Limits of Functions 199

*CHANGE OF VARIABLES IN LIMITS

"Change of variables" is a technique you have used frequently to evaluate

limits in a completely natural way. To formulate this technique formally and

justify it rigorously, however , is a task requiring some subtlety. We start with

an example. Suppose we want to obtain

. sin(~ - x)
hm ,,..
X-+~ 2 - X

We notice that if we substitute u = ~ - x, our desired limit has the form

1

. sin u
lm --.
X-+~ U

We know that lim u = lim ( !!.. - x) = 0, and we recall from calculus that

X-+~ X-+~^2

sin u

lim --= l. We thus ask whether we are justified in asserting that

u-+O u

sin ( !!.. - x) sin u

lim^2 = lim --= l.

X-+ ~ ~ - X u-+0 U

Now, let us pose the problem in a more general framework. Suppose we
want to obtain

X_,.XQ lim f (g(x))

at some cluster point xo of V(g). Suppose we know that lim g(x) = u 0 and
X-+XQ
lim f(u) = L , and we want to know whether we are justified in asserting that
u--+uo

lim f (g(x)) = li m f(u).

X--+Xo U-+Uo

The answer is "Yes,'' subject to certain technical conditions. The following
theorem provides these conditions and the justification.

•An asterisk with a theorem, proof, or other item in this chapter indicates that the item is
optional and can be omitted, especially in a one-semester course.