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There is an aspect about limits that is still confusing to me. When my class was talking about cases where limits don't exist, The teacher mentioned there are 3 and exactly 3 cases where a limit is undefined.

$\lim_{x \to c}$ is undefined when...

  1. $f(x)$ has an asymptote at $c$
  2. $\lim_{x \to c^+}\neq \lim_{x \to c^-}$. the limit approaching one side is not euqal to the limit approaching the other side, and there is a jump discontinuity at $c$.
  3. $f(x)$ is an oscillating function at $c$.

That last one the teacher wasn't really able to explain because the bell rang. So we all left rather confused, as none of us have any idea what an "oscillating function" is. Can you not calculate a limit on a function like a sine or cosine? I get Tangent because there's an asymptote that happens every $\pi$. Could she have been referencing how those functions will never come to a single value as $x$ approaches $\infty$, just bouncing between $1$ and $-1$ forever.

So... what is an oscillating function? and are Limits actually hard to calculate on sinusoidal functions?

Alexander Day
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1 Answers1

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Yes, that is exactly what she was referring to.

It doesn't just happen towards $\infty$, though. It can happen at finite points as well. Consider, for instance, $$ f(x)=\sin(1/x) $$ If you haven't seen before what its graph looks like, then I suggest you take a look, as it is a standard example of many kinds of bad behaviours that functions can have. This function doesn't have a limit as $x\to 0$ since it just oscillates more and more wildly between $-1$ and $1$.

Arthur
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