Is $\displaystyle\lim_{x\rightarrow y}\dfrac{\cos x-\cos y}{x^2-y^2}$ equal to $-\dfrac 12$ or just $-\dfrac{\sin y}{2y}$

Question

Question: $$\lim_{x\rightarrow y}\dfrac{\cos x-\cos y}{x^2-y^2}=?$$ Here is my try: \begin{align}\lim_{x\rightarrow y}\dfrac{\cos x-\cos y}{x^2-y^2}&=\lim_{x\rightarrow y}\dfrac{-2 \sin (\frac 12(x+y))\sin (\frac{1}{2}(x-y))}{(x+y)(x-y)}\\ &=-\dfrac{2 \sin y}{2y}\dfrac 12\\ &=-\dfrac{\sin y}{2y}\end{align}

My question: Is $-\dfrac{\sin y}{2y}$ the final answer or can it be calculated further as $-\dfrac12$?

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I also try different route: Let $p=x-y$ so $x=p+y$ and $p\rightarrow 0$ as $x \rightarrow y$. Hence,

\begin{align}\lim_{x\rightarrow y}\dfrac{\cos x-\cos y}{x^2-y^2}&=\lim_{p\rightarrow 0}\dfrac{\cos (p+y)-\cos y}{p^2+2py+y^2-y^2}\\\\ &=\lim_{p\rightarrow 0}\dfrac{-2 \sin (\frac{1}{2}(p+2y))\sin (\frac{1}{2}p)}{p(p+2y)}\\\\ &=\lim_{p\rightarrow 0}\dfrac{-2 \sin (\frac{1}{2}(p+2y))}{(p+2y)} \dfrac{\sin (\frac{1}{2}p)}{p}\\\\ &=-\dfrac{2\sin y}{2y} \dfrac{1}{2}\\\\ &=-\dfrac{\sin y}{2y}\end{align}

Okay, so that left me with the same result. What is the correct final answer, $-\dfrac{\sin y}{2y}$ or $-\dfrac 12$?

Thanks

Answer 1

You could also have used Taylor series around $x=y$. This is $$\cos(x)=\cos (y)-(x-y) \sin (y)-\frac{1}{2} (x-y)^2 \cos (y)+O\left((x-y)^3\right)$$ Then $$\cos(x)-\cos (y)=-(x-y) \sin (y)-\frac{1}{2} (x-y)^2 \cos (y)+O\left((x-y)^3\right)$$ $$\frac{\cos(x)-\cos (y) }{x^2-y^2}=-\frac{\sin (y)}{2 y}+(x-y) \left(\frac{\sin (y)}{4 y^2}-\frac{\cos (y)}{4 y}\right)+O\left((x-y)^2\right)$$ which shows the limit and also how it is approached.

Answer 2

Your answer is correct. A simpler approach: let $f(x)= \cos x$. Then

$\dfrac{\cos x-\cos y}{x^2-y^2}=\dfrac{f( x)-f(y)}{x-y}* \dfrac{1}{x+y} \to f'(y)* \dfrac{1}{2y}$ for $x \to y$.

Answer 3

we know by definition that $$\lim_{x\rightarrow y}\left(\frac{f(x) - f(y)}{x - y}\right) = \frac{df}{dy}$$ so the given can be expressed as $$\lim_{x\rightarrow y}\left(\frac{\cos(x) - \cos(y)}{(x - y)(x + y)}\right)$$ that is equal to $$\left(\frac{df}{dy}\right)\left(\lim_{x\rightarrow y}\frac{1}{x + y}\right)$$ at last becomes equal to $$\frac{- sin(y)}{2y}$$