3 votes
Accepted

$f$ with an essential singularity at $0$ intersects $1/z^4$ in every neighborhood of $0$.

$z$ is an element of the image of $g$, it may not even be near $0$. Nonetheless, the proof is still not very difficult. As $\dfrac{1}{z^4}$ has a pole at $0$, $f(z)-\dfrac{1}{z^4}$ has an essential ...
Julio Puerta's user avatar
  • 4,234
2 votes
Accepted

Is this function differentiable? (Error in the notes?)

This function $f$ is only defined on $[0,\infty)\times\Bbb R$. If we define differentiability of $f$ at $(0,0)$ (which is only on the boundary) by mimicking the usual definition for points interior to ...
Anne Bauval's user avatar
  • 35.2k
2 votes
Accepted

Any quicker method to evaluate $\lim_{x \to 0} \left (\frac{1+\sin x \cos \alpha x}{1+\sin x\cos\beta x}\right)^{\cot^3x}$?

I'm following up to @Claude Leibovici's answer, but using a bit different method. $$\lim_{x→0} \log(A)=\lim_{x→0} \cot ^ 3 (x)\log\left (\frac{1 + \sin(x) \cos(\alpha x)}{1 + \sin(x)\cos(\beta x)}\...
Gwen's user avatar
  • 1,028
1 vote

Any quicker method to evaluate $\lim_{x \to 0} \left (\frac{1+\sin x \cos \alpha x}{1+\sin x\cos\beta x}\right)^{\cot^3x}$?

Hints $$A= \left (\frac{1 + \sin(x) \cos(\alpha x)}{1 + \sin(x)\cos(\beta x)}\right) ^ {\cot ^ 3 (x)}$$ $$\log(A)=\cot ^ 3 (x)\log\left (\frac{1 + \sin(x) \cos(\alpha x)}{1 + \sin(x)\cos(\beta x)}\...
Claude Leibovici's user avatar

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