If we have established that $\lim_{x\to c} \sin x = \sin c$, is it enough to argue that $\cos x$ is just a translation of $\sin x$ in order to establish that $\lim_{x\to c} \cos x = \cos c$?
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1What do you think? – Parcly Taxel Oct 29 '18 at 23:10
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3I think that's right. Since sin(x) is continuous for all $x\in \mathbb R$, a translation of sin(x) must also be continuous for the same domain – marzano Oct 29 '18 at 23:12
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A translation is a continuous operation, and compositions of continuous functions are continuous. – MSDG Oct 29 '18 at 23:12
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Either you know $lim_{x\to c} f(x) = \lim_{x\to c+v} f(x - v)$ or you have to prove it. If you know it, state it. If you don't, prove it. Unless you aren't show if it's true.... in which case, figure out if it is true. – fleablood Oct 29 '18 at 23:16
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Yes since the composition of continuous function is continuous and we have
$$\cos x = \sin\left(\frac{\pi}2-x\right)$$
with $\frac{\pi}2-x$ continuous, we can conclude that $\cos x$ also is continuous.
Refer also to the related:
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