Does \lim_{(x,y)\to(3,-1)}\frac{(x-3)(y+1)}{(x-3)^2+(y+1)^2} exist?

Determine which of the following limits exist, and find the limits which do exists.
a) ${\displaystyle \underset{(x,y)\to (0,0)}{lim}\frac{x^3+y^3}{x^2+y^2}}$
b) ${\displaystyle \underset{(x,y)\to (0,0)}{lim}\frac{x^2+4x{y}^2+4y^4}{x^2+4y^4}}$

Find the limit and discuss the continuity of the function.
${\displaystyle \underset{(x,y)\to (0,1)}{lim}\frac{\arcsin xy}{1-xy}}$

Evaluate the limit
$\underset{x\to \mathrm{\infty }}{lim}\frac{4x^3-2}{3x^4+5x}$

A tank initially contains 120 L of pure water. A mixture containing a concentration of γ g/L of salt enters the tank at a rate of 2 L/min, and the well-stirred mixture leaves the tank at the same rate. Find an expression in terms of γ for the amount of salt in the tank at any time t. Also find the limiting amount of salt in the tank as ${\displaystyle t\to \mathrm{\infty }}$

Limit ${\displaystyle \underset{x\to 0}{lim}\left\{\frac{2x-\sin \left\{x\right\}}{3x+\sin \left\{x\right\}}\right\}}$

Determine the one-sided limits numerically or graphically. If infinite, state whether the one-sided limits are ${\displaystyle \mathrm{\infty }}$ or ${\displaystyle -\mathrm{\infty }}$, and describe the corresponding vertical asymptote.
${\displaystyle \underset{x\to 0^{\pm }}{lim}\frac{\sin x}{\left|x\right|}}$