March 14th, 2018, 06:34 AM  #1 
Senior Member Joined: Dec 2015 From: Earth Posts: 224 Thanks: 26  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: 376 Thanks: 205 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$. 

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