To Accumulate a Rate — Integrate!

Teaching High School Mathematics

A LaTex Sandwich

A close variation of \lim _{x\rightarrow 0} x^{4}\sin x = appeared on the AP calculus email list recently and so I thought I’d try to work up the solution here. I’ve used LaTex before and have really enjoyed using it, but sometimes it’s hard to know what to type to get the exact math typesetting your looking for. While I was typing this post I posted with one equation to see what it looked like. I came back later and I had a comment from CrazedMummy with a link to an awesome resource for LaTex code. you just draw the math equation and wahla, it gives you the code. WebEquation is great! Thank you CrazedMummy, I will use this a ton. It has now already been well worth it to start blogging. Thanks Sam.

On to the problem…

\lim _{x\rightarrow 0} x^{4}\sin x =

In order to solve this problem we need to start with the idea that the sine of anything is between or equal to one and negative one.

-1\leq \sin x \leq 1

Then we need to multiply all three parts by x^4 .
Because x^4 is always positive or zero, multiplying by one and negative one keep it the same except the negative, and since sine is always between negative one and one, multiplying x^4 by \sin x will keep it between -x^4 and x^4 .

-x^{4}\leq x^{4}\sin x \leq x^{4}

Graphically this looks like:

20120829-190248.jpg
The equation y=\sin x in red is sandwiched between y=-x^4 and y=x^4 .
This is why this is called the sandwich theorem.

You then take the limit as x goes to zero of all parts.
\lim _{x\rightarrow 0}-x^{4}\leq \lim _{x\rightarrow 0}x^{4}\sin x\leq \lim _{x\rightarrow 0}x^{4}

0\leq \lim _{x\rightarrow 0}x^{4}\sin x\leq 0

\lim _{x\rightarrow 0} x^{4}\sin x = 0 because it has to be between 0 and 0.

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7 comments on “A LaTex Sandwich

  1. crazedmummy
    August 29, 2012

    someone posted this http://webdemo.visionobjects.com/equation.html?locale=default
    and I love them for ever.

  2. denrico
    August 30, 2012

    Mmm, sandwiches. And LaTex is glorious. A LaTex sandwich, hm… not so much.

    • kaleb40
      August 31, 2012

      A LaTex Sandwich… I like it. Should have made that the post title.

      • denrico
        September 9, 2012

        Behold! I’ve made a contribution to the blogosphere! Not exactly what I’d envisioned, but a contribution nonetheless. Keep up the good work, Kaleb!

  3. denrico
    August 30, 2012

    Oh! And I can’t believe WebEquation actually understands my handwriting on a trackpad. Sick.

  4. lmhenry9
    September 8, 2012

    Neat title for your post! Nice use of both LaTeX and WebEquation.
    –Lisa

    • kaleb40
      September 9, 2012

      Thanks, I actually changed the title to this after someone said it in their comment.

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This entry was posted on August 29, 2012 by in Calculus and tagged , , , .
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