0
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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
1
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.