## Thursday, February 03, 2011

### gems from calculus

As I said last month, I'm taking a calculus class right now. Sometimes it makes me feel smart, sometimes I feel dumb. Sometimes it makes me want to cry, but every so often, the teacher come up with something that just makes me laugh. Here is today's gem (and I quote):
Have you heard of the "squeeze theorem"? No? Well, you'll find that it makes a lot of sense.
And then he proceeded to draw a graph that looked a lot like this:
And an explanation that looked a lot like this:
The limit
$\lim_{x \to 0}x^2 \sin(\tfrac{1}{x})$
cannot be ascertained through the limit law
$\lim_{x \to a}(f(x)\cdot g(x)) = \lim_{x \to a}f(x)\cdot \lim_{x \to a}g(x),$
because
$\lim_{x\to 0}\sin(\tfrac{1}{x})$
does not exist.
However, by the definition of the sine function,
$-1 \le \sin(\tfrac{1}{x}) \le 1. \,$
It follows that
$-x^2 \le x^2 \sin(\tfrac{1}{x}) \le x^2 \,$
Since $\lim_{x\to 0}-x^2 = \lim_{x\to 0}x^2 = 0$, by the squeeze theorem, $\lim_{x\to 0} x^2 \sin(\tfrac{1}{x})$ must also be 0.
(both graph and example taken from Wikipedia)

Riight...

(ok, to be fair, the theorem actually does make intuitive sense as a theorem, but it is really hard to think about demonstrating and "calculating")

My poor 30 year old brain is hurting...I'm glad I had time for a run today. And hey, who needs weight training when I had all this snow to shovel?

Hope you are all having a great week- does anyone have any fun plans for the weekend?

Laura said...

is house cleaning fun?
...then no.
Calculus? Whoooossshhh Right over me...

Suz said...

Running with you! and possibly BUNCH-rific foods :)

Kirst said...

Ooo, pick me, pick me. I for once have plans this weekend. Going to Schanks for a fundraiser for Taylor, and the kids are having a sleep over. Woohoo! PLus superbowl fun. And Chris is off. Double Woohoo! Can you tell I'm excited to have an adult night tomorrow? :)