1

I am a TA for a Calculus 1 class, and I got stumped on this question. I remember there being example cases for this when I took an Analysis class, but I can't remember any. Give an example where $$\lim_{x\to 0}(f(x)*g(x))$$ exists but neither $$\lim_{x\to 0}f(x)$$ nor $$\lim_{x\to 0}g(x)$$ exists.

Drunk Deriving
  • 387
  • $f(x)=g(x)=1/x$ – user288742 Aug 30 '16 at 15:28
  • That doesn't work. $\lim_{x\to 0}(1/x^2)=DNE$ – Drunk Deriving Aug 30 '16 at 15:30
  • $\lim \limits_{x \to 0 } 1/x^2 = \infty$ – user288742 Aug 30 '16 at 15:40
  • Notice that $f$ and $g$ must be bounded as $x \to 0$ - if $f$ were unbounded then $g$ would have to tend to $0$, which we don't want to be true. – preferred_anon Aug 30 '16 at 15:44
  • A limit that diverges to infinity fails the technical definition of a limit, epsilon-delta definition. http://math.stackexchange.com/questions/127689/why-does-an-infinite-limit-not-exist – Drunk Deriving Aug 30 '16 at 15:44
  • @MathematicsStudent there is a definition of convergence in which we can state that a function literally converges to infinity. Whether $1/x$ or $1/|x|$ or $1/x^2$ have limits at zero can become a very semantic argument. That being said, many calculus courses differentiate between $\lim_{x \to 0}f(x) = DNE$ and $\lim_{x \to 0}f(x) = +\infty$; I suggest you verify with the professor whether these situations are supposed to be distinct. – Ben Grossmann Aug 30 '16 at 15:56

3 Answers3

4

Here is an extreme example that works simultaneously at all points, not just at $0$. Let

$$f(x)=\begin{cases} 1,&\text{if }x\text{ is irrational}\\ 0,&\text{otherwise} \end{cases}$$

and

$$f(x)=\begin{cases} 1,&\text{if }x\text{ is rational}\\ 0,&\text{otherwise}\;. \end{cases}$$

Neither has a limit at any point, but their product is the continuous constant function $0$.

Brian M. Scott
  • 616,228
3

An example which might be easier for a Calc I student to understand: $$ f(x) = \begin{cases} 0 & x < 0\\ 1 & x \geq 0 \end{cases} $$ and $g(x) = 1 - f(x)$.

Ben Grossmann
  • 225,327
0

In case you don't require continuity use these $$ f(x) = \begin{cases} 0 &\mbox{, if } x =1/n \text{ for some natural n}\\ 1 & \mbox{, otherwise} \end{cases} $$

$$ g(x) = \begin{cases} 1 &\mbox{, if } x =1/n \text{ for some natural n}\\ 0 & \mbox{, otherwise} \end{cases} $$

as their product is zero everywhere.

ZuziaKru
  • 21