Two Dimensional Limit Proof Example

Question

Prove that ${\lim }_{(x,y)\to (2,1)}(x+5y)=7$ using Epsilon Delta Limit Definition

Proof

Intuition is: ${\lim }_{x\to (2,1)}(x+5y)=2+5\ast 1=7$ Given $ϵ>0$ Choose $\delta =\frac{ϵ}{6}$ Then, for any $(x,y):0<||(x,y)-(2,1)||<\delta$ $⟹|x-2|<\delta ,|y-1|<\delta$ Therefore, we hjave $|x+5y-7|=|x-2+5(y-1)|\le |x-2|+5|y-1|<\delta +5\delta =6\delta =6\ast \frac{ϵ}{6}\ast 6=ϵ$ Thus, $|x+5y-7|<ϵ⟹{\lim }_{(x,y)\to (2,1)}=7$