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Question Number 122623 by Ar Brandon last updated on 18/Nov/20
Find the limits at (0, 0) of the following functions : 1. f(x,y)=((x^2 y^2 )/(x^2 +y^2 )) 2. f(x,y)=((xy)/(x^2 +y^2 )) 3. f(x,y)=((xy)/(x+y)) 4. f(x,y)=((x^2 −y^2 )/(x^2 +y^2 )) 5. f(x,y)=((3x^2 +xy)/( (√(x^2 +y^2 )))) 6. f(x,y)=((x+y)/(x^2 +y^2 )) 7. f(x,y)=((1+x^2 +y^2 )/y)sin(y) 8. f(x,y)=((x^3 +y^3 )/(x^2 +y^2 ))
$$\mathrm{Find}\:\mathrm{the}\:\mathrm{limits}\:\mathrm{at}\:\left(\mathrm{0},\:\mathrm{0}\right)\:\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{functions}\:: \\ $$$$\mathrm{1}.\:{f}\left(\mathrm{x},\mathrm{y}\right)=\frac{\mathrm{x}^{\mathrm{2}} \mathrm{y}^{\mathrm{2}} }{\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} }\:\:\:\:\:\:\:\:\mathrm{2}.\:{f}\left(\mathrm{x},\mathrm{y}\right)=\frac{\mathrm{xy}}{\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} }\:\:\:\:\:\:\mathrm{3}.\:{f}\left(\mathrm{x},\mathrm{y}\right)=\frac{\mathrm{xy}}{\mathrm{x}+\mathrm{y}} \\ $$$$\mathrm{4}.\:{f}\left(\mathrm{x},\mathrm{y}\right)=\frac{\mathrm{x}^{\mathrm{2}} −\mathrm{y}^{\mathrm{2}} }{\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} }\:\:\:\:\:\:\:\mathrm{5}.\:{f}\left(\mathrm{x},\mathrm{y}\right)=\frac{\mathrm{3x}^{\mathrm{2}} +\mathrm{xy}}{\:\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} }}\:\:\:\:\:\:\mathrm{6}.\:{f}\left(\mathrm{x},\mathrm{y}\right)=\frac{\mathrm{x}+\mathrm{y}}{\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} } \\ $$$$\mathrm{7}.\:{f}\left(\mathrm{x},\mathrm{y}\right)=\frac{\mathrm{1}+\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} }{\mathrm{y}}\mathrm{sin}\left(\mathrm{y}\right)\:\:\:\:\mathrm{8}.\:{f}\left(\mathrm{x},\mathrm{y}\right)=\frac{\mathrm{x}^{\mathrm{3}} +\mathrm{y}^{\mathrm{3}} }{\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} } \\ $$
Answered by mathmax by abdo last updated on 18/Nov/20
1)∣xy∣≤((x^2 +y^2 )/2) ⇒ x^2 y^2 ≤(1/4)(x^2 +y^2 )^2 ⇒((x^2 y^2 )/(x^2 +y^2 ))≤(1/4)(x^2 +y^2 ) wehave lim_((x,y)→(0,0)) ((x^2 +y^2 )/4)=0 ⇒lim_((x,y)→(0,0)) f(x,y)=0 2) lim_(x→o) f(x,x) =lim_(x→0) (x^2 /(2x^2 ))=(1/2) let x=rcosθ and y=sinθ we have f(x,y) =((r^2 cosθsinθ)/r^2 )=(1/2)sin(2θ)⇒lim_((x,y)) f(x,y) dont exist..!
$$\left.\mathrm{1}\right)\mid\mathrm{xy}\mid\leqslant\frac{\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} }{\mathrm{2}}\:\Rightarrow\:\mathrm{x}^{\mathrm{2}} \mathrm{y}^{\mathrm{2}} \leqslant\frac{\mathrm{1}}{\mathrm{4}}\left(\mathrm{x}^{\mathrm{2}} \:+\mathrm{y}^{\mathrm{2}} \right)^{\mathrm{2}} \:\Rightarrow\frac{\mathrm{x}^{\mathrm{2}} \mathrm{y}^{\mathrm{2}} }{\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} }\leqslant\frac{\mathrm{1}}{\mathrm{4}}\left(\mathrm{x}^{\mathrm{2}} \:+\mathrm{y}^{\mathrm{2}} \right)\:\mathrm{wehave} \\ $$$$\mathrm{lim}_{\left(\mathrm{x},\mathrm{y}\right)\rightarrow\left(\mathrm{0},\mathrm{0}\right)} \:\:\:\frac{\mathrm{x}^{\mathrm{2}} \:+\mathrm{y}^{\mathrm{2}} }{\mathrm{4}}=\mathrm{0}\:\Rightarrow\mathrm{lim}_{\left(\mathrm{x},\mathrm{y}\right)\rightarrow\left(\mathrm{0},\mathrm{0}\right)} \:\:\mathrm{f}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{lim}_{\mathrm{x}\rightarrow\mathrm{o}} \mathrm{f}\left(\mathrm{x},\mathrm{x}\right)\:=\mathrm{lim}_{\mathrm{x}\rightarrow\mathrm{0}} \frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2x}^{\mathrm{2}} }=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\mathrm{let}\:\mathrm{x}=\mathrm{rcos}\theta\:\mathrm{and}\:\mathrm{y}=\mathrm{sin}\theta\:\:\mathrm{we}\:\mathrm{have} \\ $$$$\mathrm{f}\left(\mathrm{x},\mathrm{y}\right)\:=\frac{\mathrm{r}^{\mathrm{2}} \mathrm{cos}\theta\mathrm{sin}\theta}{\mathrm{r}^{\mathrm{2}} }=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{sin}\left(\mathrm{2}\theta\right)\Rightarrow\mathrm{lim}_{\left(\mathrm{x},\mathrm{y}\right)} \mathrm{f}\left(\mathrm{x},\mathrm{y}\right)\:\mathrm{dont}\:\mathrm{exist}..! \\ $$

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