I have a homework question to find the points at which $f(x)=\frac{{{\sin }^{2}}x}{x|x(\pi -x)|}$ is not continuous. I can't seem to figure it out for this function.
Can someone help me find these points?
Thanks!
I have a homework question to find the points at which $f(x)=\frac{{{\sin }^{2}}x}{x|x(\pi -x)|}$ is not continuous. I can't seem to figure it out for this function.
Can someone help me find these points?
Thanks!
A function $f(x)$ is continuous at $x=a$ if and only if it satisfies three conditions:
By the usual limit laws and the definitions of sums, products, quotients, and compositions of functions, we have:
Now, $f(x)=\sin(x)$ is continuous everywhere, so the product of $f$ with itself, $\sin^2(x)$, is continuous everywhere.
The constant function $f(x)=\pi$ is continuous everywhere, the identity function $g(x)=x$ is continuous everywhere, so their difference, $x\mapsto \pi - x$ is continuous everywhere.
The function $f(x)=\pi -x$ is continuous everywhere, the identity function $g(x)=x$ is continuous everywhere, so their product $fg(x) = x(\pi -x)$ is continuous everywhere.
The function $f(x)=x(\pi-x)$ is continuous everwhere, the function $g(x)=|x|$ is continuous everywhere, so their composition $g\circ f(x) = |x(\pi-x)|$ is continuous everywhere.
Similarly, the function $x\mapsto x|x(\pi - x)|$ is continuous eveywhere.
So: the numerator of your function is continuous everywhere. The denominator of your function is continuous everywhere. So... where is their quotient continuous?