Three Down One Left

Three midterms completed, one is left next week. I’ve got two major assignments due next week, but they are all but completed. Another one is due on the week of the 27th of March. I still have to start that one.

I got B in my first one midterm. I lost most of my points in the first exercise which I had trouble with. It was worth a quarter of the exam. I did the second midterm on Thursday. It went very well. My third was on Friday. It went OK. There was one exercise that I couldn’t complete. In retrospect, it was an easy one. I just needed to use some analytical tools to analyze a mathematical series. What messed me up is that this was a Numerical Methods class, not analysis. I didn’t think of using le Théorême des deux gendarmes or the Squeeze Theorem.

Squeeze Theorem

An example of a squeezed function

The squeeze theorem is formally stated as follows.

Let I be an interval containing the point a. Let f, g, and h be functions defined on I, except possibly at a itself. Suppose that for every x in I not equal to a, we have:

g(x) \leq f(x) \leq h(x)

and also suppose that:

\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L.

Then \lim_{x \to a} f(x) = L.

  • The functions g(x) and h(x) are said to be lower and upper bounds (respectively) of f(x).
  • Here a is not required to lie in the interior of I. Indeed, if a is an endpoint of I, then the above limits are left- or right-hand limits.
  • A similar statement holds for infinite intervals: for example, if I = (0, ∞), then the conclusion holds, taking the limits as x → ∞.

It’s a very powerful analytical tool to deal with infinite mathematical series, since it’s easy to major and minor them.
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