March 14th, 2018, 06:34 AM  #1 
Senior Member Joined: Dec 2015 From: somewhere Posts: 633 Thanks: 91  Equation from analysis
$\displaystyle lim_{y\rightarrow x} \frac{\sin y}{y}=1$ Find $\displaystyle x=?$ 
March 14th, 2018, 07:54 AM  #2 
Senior Member Joined: Sep 2016 From: USA Posts: 642 Thanks: 406 Math Focus: Dynamical systems, analytic function theory, numerics 
$x = 0$ is the only solution. This follows since \[\lim_{y \to x} \frac{\sin y}{y} = 1 \iff \lim_{y \to x} \sin y  y = 0 \] But since $\sin y  y$ is a continuous function, we can compute this limit explicitly as \[ \lim_{y \to x} \sin y  y = \sin x  x \] and equating the 2 values for the limit implies $x$ satisfies $\sin x = x$ whose only solution is $ x = 0$. 

Tags 
analysis, equation 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Analysis of Image superimposing using numerical analysis  saugata bose  Computer Science  0  March 8th, 2013 11:03 AM 
Understanding some items on equation of fourier analysis  nepdeep  Real Analysis  3  March 7th, 2013 08:41 PM 
Data analysis using Independent Component Analysis (ICA)  ggyyree  Advanced Statistics  1  August 9th, 2012 03:48 AM 
Prove between Real Analysis and Complex Analysis  uniquesailor  Real Analysis  2  January 3rd, 2012 09:56 PM 
Video discusses interesting new view of equation analysis  Bon Qui Qui  Number Theory  0  October 10th, 2011 12:11 AM 