Question Number 139432 by normabaru last updated on 27/Apr/21
Answered by Jme Eduardo last updated on 27/Apr/21
![(a) if x=t^2 and y=t lim_(t→0) [((2t^4 )/(t^4 +t^4 ))]= 1 if x=t and y=t^2 lim [((2t^4 )/(t^2 +t^8 ))]= lim_(t→0) [((2t)/(1+t^6 ))]= 0 the lim_((x;y)→(0;0)) f(x;y) not exist, because if we approach parametric curves the function has different limits](https://www.tinkutara.com/question/Q139440.png)
$$\left({a}\right)\:\:{if}\:\:{x}={t}^{\mathrm{2}} \:\:{and}\:\:{y}={t} \\ $$$$ \\ $$$$\underset{{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left[\frac{\mathrm{2}{t}^{\mathrm{4}} }{{t}^{\mathrm{4}} +{t}^{\mathrm{4}} }\right]=\:\mathrm{1} \\ $$$$ \\ $$$${if}\:\:{x}={t}\:\:{and}\:\:{y}={t}^{\mathrm{2}} \\ $$$$ \\ $$$$\mathrm{lim}\:\left[\frac{\mathrm{2}{t}^{\mathrm{4}} }{{t}^{\mathrm{2}} +{t}^{\mathrm{8}} }\right]=\:\underset{{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left[\frac{\mathrm{2}{t}}{\mathrm{1}+{t}^{\mathrm{6}} }\right]=\:\mathrm{0} \\ $$$$ \\ $$$${the}\:\underset{\left({x};{y}\right)\rightarrow\left(\mathrm{0};\mathrm{0}\right)} {\mathrm{lim}}\:\:\:{f}\left({x};{y}\right)\:{not}\:{exist},\:{because}\:{if}\:{we}\:{approach}\:{parametric}\:{curves}\:{the}\:{function}\:{has}\:\:{different}\:\:{limits} \\ $$
Answered by Jme Eduardo last updated on 27/Apr/21
![(a) if x=t^2 and y=t lim_(t→0) [((2t^4 )/(t^4 +t^4 ))]= 1 if x=t and y=t^2 lim [((2t^4 )/(t^2 +t^8 ))]= lim_(t→0) [((2t)/(1+t^6 ))]= 0 the lim_((x;y)→(0;0)) f(x;y) not exist, because if we approach parametric curves the function has different limits](https://www.tinkutara.com/question/Q139441.png)
$$\left({a}\right)\:\:{if}\:\:{x}={t}^{\mathrm{2}} \:\:{and}\:\:{y}={t} \\ $$$$ \\ $$$$\underset{{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left[\frac{\mathrm{2}{t}^{\mathrm{4}} }{{t}^{\mathrm{4}} +{t}^{\mathrm{4}} }\right]=\:\mathrm{1} \\ $$$$ \\ $$$${if}\:\:{x}={t}\:\:{and}\:\:{y}={t}^{\mathrm{2}} \\ $$$$ \\ $$$$\mathrm{lim}\:\left[\frac{\mathrm{2}{t}^{\mathrm{4}} }{{t}^{\mathrm{2}} +{t}^{\mathrm{8}} }\right]=\:\underset{{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left[\frac{\mathrm{2}{t}}{\mathrm{1}+{t}^{\mathrm{6}} }\right]=\:\mathrm{0} \\ $$$$ \\ $$$${the}\:\underset{\left({x};{y}\right)\rightarrow\left(\mathrm{0};\mathrm{0}\right)} {\mathrm{lim}}\:\:\:{f}\left({x};{y}\right)\:{not}\:{exist},\:{because}\:{if}\:{we}\:{approach}\:{parametric}\:{curves}\:{the}\:{function}\:{has}\:\:{different}\:\:{limits} \\ $$