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March 14th, 2018, 06:34 AM   #1
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Equation from analysis

$\displaystyle lim_{y\rightarrow x} \frac{\sin y}{y}=1$
Find $\displaystyle x=?$
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March 14th, 2018, 07:54 AM   #2
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$x = 0$ 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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