I have the following exercise:
Consider the function $$f : \mathbb R \times \mathbb R^+ \to \mathbb R \,,\ f(x,y) := \frac{\log\left(1+(y-x)^2\right)}{y^2+y}$$
- Does there exist the limit of $f(x,y)$ for $(x,y) \to (0,0)$? If we restrict to $A := \left\{ (x, y) \in \mathbb R \times \mathbb R^+ \mid y \ge \lvert x \rvert\right\}$, is there a limit?
- Let $\chi_n : \mathbb R^2 \to \mathbb R$ that maps $(x,y)$ to $1$ if $0 \le x^2+y^2 \le \frac{1}{n^2}$, otherwise to $0$. Study the pointwise and uniform convergence of the sequence of functions $$g_n : \mathbb R \times \mathbb R^+ \to \mathbb R \,,\ g_n(x,y) := f(x,y) \left(1-\chi_n(x,y)\right)$$
Here is my solution.
Let $x = \rho \cos \theta$ and $y = \rho \sin \theta$. If $y \ge \lvert x \rvert$, then $\sin \theta \ge \frac1{\sqrt2}$. $$\lvert f(\rho \cos \theta, \rho \sin \theta)\rvert \le\frac{\rho^2}{\rho \sin \theta}$$ Consequently, $$\lvert f(x,y)\rvert \le \sqrt2 \rho$$ and $f(x,y) \to 0$ if $(x,y) \in A$ and $(x,y) \to (0,0)$.
Over the whole $\mathbb R \times \mathbb R^+$ we cannot find the limit. For example $$f\left(x, x^2\right) \sim \frac{\left(x^2-x\right)^2}{x^4+x^2} \to 1 \quad \text{for } x \to 0$$
The pointwise convergence is simple: for every $(x,y) \in \mathbb R \times \mathbb R^+$ we have $g_n(x,y) = f(x,y)$ definitively for $n \to \infty$. Hence $g_n \to f$ pointwise. If $(x,y) \in A$, then $$\left\lvert g_n(x,y)-f(x,y) \right\rvert = \chi_n(x,y) \lvert f(x,y) \rvert \le \frac{\sqrt 2}n$$ I can conclude that $g_n \to f$ uniformly over $A$ too. In general, that is not the case though: for example $\left\lvert f\left(x, x^2\right)\right\rvert \ge \frac12$ in some neigbourdhood of $(0,0)$.
Is my solution correct?