Show that the function $f(x)= |\sin(x)+\cos(x)|$ is continuous at $x=\pi$.
By drawing the graph, we can easily show that it is continuous, but how can we show it by using limits. Please help.
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Can you show us what you have tried? It is ggod to show what you have tried so far to get good responses. – lsp Dec 09 '14 at 10:50
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How about as a composite of continuous function? – Olivier Bégassat Dec 09 '14 at 10:51
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2$\sin x$, $\cos x$ and $\left| x \right|$ are all continuous. So $f(x)$ is continuous. – peterwhy Dec 09 '14 at 10:53
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The number defined as the ratio between the diameter and circumference of a circle is called $\pi$, a greek letter which is spelled pi, often pronounced as pie, but please don't spell it like that. – Henrik supports the community Dec 09 '14 at 10:54
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This can be proven in a number of ways. Firstly as Olivier Begassat noted that it is a composition of continues functions and thus is continues itself.
Another way you can see it is by observing that:
$$\lim_{x\downarrow \pi} f(x)=|\sin(\pi)+\cos(\pi)|=\lim_{x\uparrow \pi}f(x)=\lim_{x\to \pi}f(x)=1$$ So the function exists and is well defined in the limit. Therefore it is conitnues