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 Calculus Calculus Math Forum

 July 8th, 2019, 05:44 AM #1 Senior Member   Joined: Oct 2015 From: Greece Posts: 137 Thanks: 8 Sandwich Theorem Trig Funcs. For some reason I struggle using the sandwich theorem in solving trigonometric function limits. Example 1.png Solution 2.png But I can not understand how he does that. My try using $\displaystyle -|θ| < sinθ < |θ|$ : $\displaystyle -|2x| < sin2x < |2x| \Leftrightarrow \frac{-|2x|}{x} < \frac{sin2x}{x} < \frac{|2x|}{x}$ what now? how does the books reaches this: $\displaystyle \frac{-1}{x} < \frac{sin2x}{x} < \frac{1}{x}$ ???? How does he also get rid of the absolute value? I solved this with words $\displaystyle \lim_{x \to \infty} \frac{sin2x}{x}$, $\displaystyle \lim_{x \to \infty} sin2x = a \epsilon R$, so $\displaystyle \frac{a}{\lim_{x \to \infty}x}=0$ Last edited by babaliaris; July 8th, 2019 at 05:50 AM. July 8th, 2019, 06:13 AM #2 Math Team   Joined: Jul 2011 From: Texas Posts: 3,016 Thanks: 1600 the value of sin(any angle) is inclusively between -1 and 1, i.e $-1 \le \sin(2x) \le 1$ since $x \to \infty$, $x$ is a positive, non-zero number and one can divide each term in the above inequality by $x$ $\dfrac{-1}{x} \le \dfrac{\sin(2x)}{x} \le \dfrac{1}{x}$ as $x \to \infty$, both $-\dfrac{1}{x}$ and $\dfrac{1}{x}$ tend to zero ... Thanks from babaliaris July 8th, 2019, 06:18 AM   #3
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 Originally Posted by skeeter the value of sin(any angle) is inclusively between -1 and 1, i.e $-1 \le \sin(2x) \le 1$ since $x \to \infty$, $x$ is a positive, non-zero number and one can divide each term in the above inequality by $x$ $\dfrac{-1}{x} \le \dfrac{\sin(2x)}{x} \le \dfrac{1}{x}$ as $x \to \infty$, both $-\dfrac{1}{x}$ and $\dfrac{1}{x}$ tend to zero ...
When can this inequality $\displaystyle -|θ| < sinθ < |θ|$ be useful? Tags funcs, sandwich, theorem, trig Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Basko Algebra 4 May 15th, 2019 03:45 PM 123qwerty Pre-Calculus 13 December 27th, 2015 08:27 AM leo255 Calculus 1 November 10th, 2014 01:46 PM James1973 Advanced Statistics 0 October 14th, 2013 02:42 AM cr1pt0 Trigonometry 2 September 5th, 2013 06:11 PM

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