Squeeze theorem: Difference between revisions

In [[calculus]], the '''squeeze theorem''', (also known as the '''pinchingsandwich theorem''', among other names{{efn|Also known as the '''sandwichpinching theorem''', the '''sandwich rule'', the ''police theorem'', the ''between theorem'' and sometimes the '''squeeze lemma'''. In Italy, the theorem is also known as the ''theorem of carabinieri''.}}) is a [[theorem]] regarding the [[limit of a function|limit]]. Inof Italy,a the[[Function theorem(mathematics)|function]] that is alsobounded knownbetween astwo '''theoremother of Carabinieri'''functions.

The squeeze theorem is used in calculus and [[mathematical analysis]]. It is, typically used to confirm the [[limit]] of a function via comparison with two other functions whose limits are known or easily computed. It was first used geometrically by the [[mathematician]]smathematicians [[Archimedes]] and [[Eudoxus of Cnidus|Eudoxus]] in an effort to compute [[pi|{{pi}}]], and was formulated in modern terms by [[Carl Friedrich Gauss]].

The squeeze theorem is formally stated as follows.<ref>{{cite book|last1=Sohrab|first1=Houshang H.|title=Basic Real Analysis| date=2003|publisher=[[Birkhäuser]]|isbn=978-1-4939-1840-9|page=104|edition=2nd|url=https://books.google.com/books?id=QnpqBQAAQBAJ&pg=PA104}}</ref>

Let ''{{mvar|I''}} be an [[interval (mathematics)|interval]] havingcontaining the point ''{{mvar|a'' as a limit point}}. Let ''{{mvar|g''}}, ''{{mvar|f''}}, and ''{{mvar|h''}} be [[function (mathematics)|functions]] defined on ''{{mvar|I''}}, except possibly at ''{{mvar|a''}} itself. Suppose that for every ''{{mvar|x''}} in ''{{mvar|I''}} not equal to ''{{mvar|a''}}, we have

* The functions <math display="inline">{{mvar|g</math>}} and <math display="inline">{{mvar|h</math>}} are said to be [[upper bound|lower and upper bounds]] (respectively) of <math display="inline">{{mvar|f</math>}}.

* Here, <math display="inline">{{mvar|a</math>}} is ''not'' required to lie in the [[interior (topology)|interior]] of <math display="inline">{{mvar|I</math>}}. Indeed, if <math display="inline">{{mvar|a</math>}} is an endpoint of <math display="inline">{{mvar|I</math>}}, then the above limits are left- or right-hand limits.

* A similar statement holds for infinite intervals: for example, if <{{math display|1="inline">''I'' = (0, \infty&infin;)</math>}}, then the conclusion holds, taking the limits as <{{math display="inline">|''x'' \rightarrow→ \infty</math>&infin;}}.

This theorem is also valid for sequences. Let <{{math>|(a_n''a{{sub|n}}''), (c_n''c{{sub|n}}'')</math>}} be two sequences converging to <math>\ell</math>{{mvar|ℓ}}, and <{{math>|(b_n''b{{sub|n}}'')</math>}} a sequence. If <math>\forall n\geqslantgeq N, N\in\mathbb{N}</math> we have <{{math>a_n\leqslant|''a{{sub|n}}'' b_n\leqslant≤ c_n</math>''b{{sub|n}}'' ≤ ''c{{sub|n}}''}}, then <{{math>|(b_n''b{{sub|n}}'')</math>}} also converges to <math>\ell</math>{{mvar|ℓ}}.

FromAccording to the above hypotheses we have, taking the [[limit inferior]] and superior:

:<math display="block">L=\lim_{x \to a} g(x)\leq\liminf_{x\to a}f(x) \leq \limsup_{x\to a}f(x)\leq \lim_{x \to a}h(x)=L,</math>

A direct proof, using the <{{math>|(\''&epsilon;'', \''&delta;'')</math>}}-definition of limit, would be to prove that for all real <{{math display="inline">\|''&epsilon;'' > 0</math>}} there exists a real <{{math>\|''&delta;'' > 0</math>}} such that for all <math>{{mvar|x</math>}} with <math>|x - a| < \delta,</math>, we have <math>|f(x) - L| < \epsilonvarepsilon.</math>. Symbolically,

: <math display="block"> \forall \epsilonvarepsilon > 0, \exists \delta > 0 : \forall x, (|x - a | < \delta \ \Rightarrow |f(x) - L |< \epsilonvarepsilon).</math>

: <math display="block">\lim_{x \to a} g(x) = L </math>

This means that

: <math display="block">\lim_{x \to a} h(x) = L </math>

: <math display="block">g(x) \leq f(x) \leq h(x) </math>

: <math display="block">g(x) - L\leq f(x) - L\leq h(x) - L</math>

We can choose <math>\delta:=\min\left\{\delta_1,\delta_2\right\}</math>. Then, if <math>|x - a| < \delta</math>, combining ({{EquationNote|1}}) and ({{EquationNote|2}}), we have

: <math display="block"> - \varepsilon < g(x) - L\leq f(x) - L\leq h(x) - L\ < \varepsilon, </math>

: <math display="block"> - \varepsilon < f(x) - L < \varepsilon ,</math>,

which completes the proof. <math>\blacksquare</math>[[Q.E.D]]

The proof for sequences is very similar, using the <math>\epsilonvarepsilon</math>-definition of athe limit of a sequence.

[[File:Inst_satsen.png|thumb|right|250px|''x''<supmath>x^2</sup>&nbsp; \sin\left(\tfrac{1/''}{x''}\right)</math> being squeezed in the limit as {{mvar|x}} goes to 0]]

: <math display="block">\lim_{x \to 0}x^2 \sin\left( \tfrac{1}{x} \right)</math>

: <math display="block">\lim_{x \to a}(f(x) \cdot g(x)) =

\lim_{x \to a}f(x) \cdot \lim_{x \to a}g(x),</math>

: <math display="block">\lim_{x\to 0}\sin\left( \tfrac{1}{x} \right)</math>

: <math display="block">-1 \le \sin\left( \tfrac{1}{x} \right) \le 1. </math>

: <math display="block">-x^2 \le x^2 \sin\left( \tfrac{1}{x} \right) \le x^2 </math>

Since <math>\lim_{x\to 0}-x^2 = \lim_{x\to 0}x^2 = 0</math>, by the squeeze theorem, <math>\lim_{x\to 0} x^2 \sin\left(\tfrac{1}{x}\right)</math> must also be 0.

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

& \lim_{x\to 0} \frac{\sin( x)}{x} =1, \\[10pt]

& \lim_{x\to 0} \frac{1 - \cos( x)}{x} = 0.

: <math display="block"> \cos x \leq \frac{\sin( x)}{x} \leq 1 </math><ref>Selim G. Krejn, V.N. Uschakowa: ''Vorstufe zur höheren Mathematik''. Springer, 2013, {{ISBN|9783322986283}}, pp.

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)</math>''}} in the earlier fact by <math display="inline"> \sqrt{1-\cos(x)^2 x}</math> and squaring the resulting inequality.

: <math display="block"> \frac{\sec^2\theta\,\Delta\theta}{2}, </math>

since the radius is {{math|sec&nbsp; ''&theta;θ''}} and the arc on the [[unit circle]] has length&nbsp;{{math|Δ''&theta;θ''}}. Similarly, the area of the larger of the two shaded sectors is

: <math display="block"> \frac{\sec^2(\theta + \Delta\theta)\,\Delta\theta}{2}. </math>

What is squeezed between them is the triangle whose base is the vertical segment whose endpoints are the two dots. The length of the base of the triangle is&nbsp; {{math|tan(''&theta;θ''&nbsp; +&nbsp; Δ''&theta;θ'')&nbsp;&minus;&nbsp; − tan( ''&theta;θ'')}}, and the height is&nbsp;1. The area of the triangle is therefore

: <math display="block"> \frac{\tan(\theta + \Delta\theta) - \tan(\theta)}{2}. </math>

: <math display="block"> \frac{\sec^2\theta\,\Delta\theta}{2} \le \frac{\tan(\theta + \Delta\theta) - \tan(\theta)}{2} \le \frac{\sec^2(\theta + \Delta\theta)\,\Delta\theta}{2} </math>

: <math display="block"> \sec^2\theta \le \frac{\tan(\theta + \Delta\theta) - \tan(\theta)}{\Delta\theta} \le \sec^2(\theta + \Delta\theta),</math>

provided&nbsp;{{math|Δ''&theta;θ''&nbsp; >&nbsp; 0}}, and the inequalities are reversed if&nbsp;{{math|Δ''&theta;θ''&nbsp; <&nbsp; 0}}. Since the first and third expressions approach {{math|sec²''&theta;θ''}} as {{math|Δ''&theta;θ''&nbsp; →&nbsp; 0}}, and the middle expression approaches (''<math>\tfrac{d''/''}{d&\theta;'')&nbsp;} \tan&nbsp;''&\theta;'',</math> the desired result follows.

The squeeze theorem can still be used in multivariable calculus but the lower (and upper functions) must be below (and above) the target function not just along a path but around the entire neighborhood of the point of interest and it only works if the function really does have a limit there. It can, therefore, be used to prove that a function has a limit at a point, but it can never be used to prove that a function does not have a limit at a point.<ref>{{cite book|chapter=Chapter 15.2 Limits and Continuity| pages=909–910|title=Multivariable Calculus|year=2008|last1=Stewart|first1=James|authorlink1 author-link1=James_Stewart_James Stewart (mathematician)| edition=6th|isbn=0495011630978-0495011637}}</ref>

: <math display="block">\lim_{(x,y) \to (0, 0)} \frac{x^2 y}{x^2+y^2}</math>

: <math>-\left |& y0 & \rightleq & \vertdisplaystyle \leqfrac{x^2}{x^2+y^2} y& \leq \left& | y1 \right \vert </math>[4pt]

: <math>-\left -|y| \leq y \rightleq |y| \vertimplies & -|y| & \leq & \displaystyle \frac{x^2 y}{x^2+y^2} & \leq \left& | y| \right \vert </math>[4pt]

: <math> {\displaystyle \lim_{(x,y) \to (0, 0)} \left -|y \right \vert| = 0</math>} \atop

: <math>0 {\leq displaystyle \lim_{(x,y) \to (0, 0)} \frac{x^2 y}{x^2+y^2} \ \leq |y| = 0</math>}

: <math display="block">\lim_{(x,y) \to (0, 0)} \frac{x^2 y}{x^2+y^2} = 0.</math>

[[Category:Theorems inabout calculusreal number sequences]]