Editing Squeeze theorem (section)

=== Second example ===
[[File:Limit_sin_x_x.svg|thumb|upright=1.5|Comparing areas:<br/>
<math>\begin{array}{cccccc}
  & A(\triangle ADB) & \leq & A(\text{sector } ADB) & \leq & A(\triangle ADF) \\[4pt]
  \Rightarrow & \frac{1}{2} \cdot \sin x \cdot 1 & \leq & \frac{x}{2\pi} \cdot \pi & \leq & \frac{1}{2} \cdot \tan x \cdot 1 \\[4pt] 
  \Rightarrow & \sin x & \leq & x & \leq & \frac{\sin x}{\cos x} \\[4pt]
  \Rightarrow & \frac{\cos x}{\sin x} & \leq & \frac{1}{x} & \leq & \frac{1}{\sin x} \\[4pt]
  \Rightarrow & \cos x & \leq & \frac{\sin x}{x} & \leq & 1 
  \end{array}</math>]] 

Probably the best-known examples of finding a limit by squeezing are the proofs of the equalities
<math display="block">
\begin{align}
& \lim_{x\to 0} \frac{\sin x}{x} =1, \\[10pt]
& \lim_{x\to 0} \frac{1 - \cos x}{x} = 0.
\end{align}
</math>

The first limit follows by means of the squeeze theorem from the fact that<ref>Selim G. Krejn, V.N. Uschakowa: ''Vorstufe zur höheren Mathematik''. Springer, 2013, {{ISBN|9783322986283}}, pp. [https://books.google.com/books?id=-yXMBgAAQBAJ&pg=PA80 80-81] (German). See also [[Sal Khan]]: [https://www.khanacademy.org/math/ap-calculus-ab/limits-from-equations-ab/squeeze-theorem-ab/v/proof-lim-sin-x-x ''Proof: limit of (sin x)/x at x=0''] (video, [[Khan Academy]])</ref>

<math display="block"> \cos x \leq \frac{\sin x}{x} \leq 1 </math>

for {{mvar|x}} close enough to 0. The correctness of which for positive {{mvar|x}} can be seen by simple geometric reasoning (see drawing) that can be extended to negative {{mvar|x}} as well. The second limit follows from the squeeze theorem and the fact that

<math display="block"> 0 \leq  \frac{1 - \cos x}{x}  \leq  x </math>
for {{mvar|x}} close enough to 0. This can be derived by replacing {{math|sin ''x''}} in the earlier fact by <math display="inline"> \sqrt{1-\cos^2 x}</math> and squaring the resulting inequality.

These two limits are used in proofs of the fact that the derivative of the sine function is the cosine function.  That fact is relied on in other proofs of derivatives of trigonometric functions.