3
$\begingroup$

The book says that $\lim_{x \rightarrow 0^{-}} \left( \frac{1}{x} - \frac{1}{|x|} \right) \mbox{does not exist}$
But, given any $M \lt 0$ of large magnitude, if I choose $\delta = \frac{-x^{2}M}{2}$ then any value of x where $|x-0|< \delta$ and $x <0$ (as we are coming from the left) will lead to $\left( \frac{1}{x} - \frac{1}{|x|} \right) < M$. To me, that says that my text book is incorrect in saying that this limit "d.n.e."

I'm a little bothered that my $\delta$ depends on $x$, but I tried a few numerical examples and it worked fine. Perhaps the function is not uniformly continuous when $x \lt 0$? I have not done enough work to answer that question yet.

Maybe the book meant to say
$\lim_{x \rightarrow 0} \left( \frac{1}{x} - \frac{1}{|x|} \right) \mbox{does not exist?}$ Or maybe I have missed something elementary.

  • 0
    @a little don: $\delta$ should not depend on $x$; it seems you might be confusing$a$bit with uniform continuity. In general, to show$a$function is continuous at $a$ your $\delta$ may depend on $a$ and$\epsilon$(but not on $x$); the function is uniformly continuous if $\delta$ does not depend **on $a$**.2011-02-11

1 Answers 1

4

One problem with your $\delta$ is that $|x|\lt \frac{-x^2M}{2}$ with $x\neq0$ implies $|x|\gt-\frac{2}{M}$. Having this positive lower bound on $|x|$ means that you are not actually approaching $0$. As Jason DeVito indicates, taking advantage of the fact that $x\lt0$ to write $|x|=-x$ makes it easier to see why the limit doesn't exist.

In a comment you just mentioned that the limit is $-\infty$. That is true, but then there is a divide in terminology as to what having a limit means. Infinite limits can be dealt with in terms of convergence in the extended reals, $[-\infty,+\infty]$, but often $\pm\infty$ are treated separately with respect to limits, and the existence of a limit (without further qualification) often only refers to existence of a limit in the real number system, $(-\infty,+\infty)$.

  • 3
    @a little don: in most of the calculus texts that I've seen, they will write that a limit "equals infinity" (or negative infinity) as a shorthand for saying that the limit does not exist, but does not exist in a specific way that is further described as the expression increasing (or decreasing) without bound.2011-02-11