Show that the following function is continuous on R^2

Question

Consider the function f : $\displaystyle\mathbb{R^2}→ \mathbb{R}$ with $$f(x_1,x_2)=\begin{cases} \frac{sin(|x_2|)cot(|x_1|)+cos(|x_2|)}{sin(|x_2|)-cot(|x_1|)cos(|x_2|)}(|x_1|+|x_2|)^{-1}, & \text{if $(x_1,x_2)\neq(0,0)$} \\ 0, & \text{if $(x_1,x_2)=(0,0)$} \end{cases}$$ Is the function continuous on $\displaystyle\mathbb{R^2}$? $$$$ I tried to rewrite the function $f(x_1,x_2)$ as a function only depending on a norm of $(x_1,x_2)$, and I also tried to show if it is continuous by using a sequence $x_k=(\frac{1}{k},\frac{1}{k})$, which converges to 0, and replace it to $f(x_1,x_2)$ but still I can't get anywhere with that and to be honest I don't know how to continue anymore. Any solution is appreciated.

Answer 1

You do the substitution $x=|x_1|+|x_2|$. The fraction of the trigonometric functions on the left side simplifies drastically if you replace the cotagent by a fraction of sine and cosine, and then use the addition theorems. In the end you get $-\tan(x)/x$. This converges to $-1$.