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.

  • $\begingroup$ Can you show us what you have tried? It is ggod to show what you have tried so far to get good responses. $\endgroup$ – lsp Dec 9 '14 at 10:50
  • $\begingroup$ How about as a composite of continuous function? $\endgroup$ – Olivier Bégassat Dec 9 '14 at 10:51
  • 2
    $\begingroup$ $\sin x$, $\cos x$ and $\left| x \right|$ are all continuous. So $f(x)$ is continuous. $\endgroup$ – peterwhy Dec 9 '14 at 10:53
  • $\begingroup$ 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. $\endgroup$ – Henrik - stop hurting Monica Dec 9 '14 at 10:54

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


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