How do I find the limit as $(x,y) \rightarrow (0,0)$ for $\frac{\cos(x)+\cos(y)-2}{\sqrt{x^2+y^2}}$ and $\frac{\sin(x+2y)-x-2y}{\sqrt{x^2+y^2}}$
2026-05-15 07:22:06.1778829726
Finding limits in two variables with trig functions
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We have:$$\cos x+\cos y-2=-4(\sin^2 {x\over 2}+\sin^2 {y\over 2})\sim -2x^2-2y^2$$therefore the first limit becomes zero as well as the 2nd one which becomes zero either using Taylor series of $\sin u$ in $u=0$.