<p>Can someone explain to me the proof for the pinching theorem?
Not just what to do, but also why certain steps are taken, etc.?</p>
<p>I would appreciate it if you could use h(x)≤f(x)≤g(x) as the three functions.</p>
<p>Did this come from your teacher? Because in order to formally prove the theorem and the Limit Laws you need to understand what the delta-epsilon definition of limits is, which is not tested in the exam b/c it is a hard concept. Understanding the theorem is already enough because the AP exam will ask you to justify what you are doing when you answer the FRQs but not to prove theorems.</p>
<p>Wikipedia has a simple proof of the squeeze theorem compared to one probably found in your textbook. Suppose h(x)≤f(x)≤g(x) and you take the limit superior and limit inferior of f(x) (the upper limit bound and lower limit bound, respectively). If the upper limit bound is less than or equal to the limit of g(x), AND the lower limit bound is more than or equal to the limit of h(x), then the squeeze theorem holds (applies here) because g(x) is also an upper bound and h(x) is also an lower bound. So even though f(x) doesn’t have to equal h(x) and/or g(x), its limit will be squeezed between the limit of h(x) and g(x) so all three limits are equal. I hope that helps.</p>
<p>khan academy has a great video that will exactly answer your question.</p>