\(\displaystyle\lim_{{{\left({x},{y}\right)}\rightarrow{\left({0},{0}\right)}}}{\left(-\frac{{{x}{y}}}{{{x}^{{2}}+{y}^{{2}}}}\right)}\)

Find two dofferent paths to approach the point that gives different values for

Limit of \(\displaystyle\lim_{{{\left({x},{y}\right)}\rightarrow{\left({0},{0}\right)}}}{\left(-\frac{{{x}{y}}}{{{x}^{{2}}+{y}^{{2}}}}\right)}{a}{l}{o}{n}{g}{x}={0}:{0}\)

Limit of \(\displaystyle\lim_{{{\left({x},{y}\right)}\rightarrow{\left({0},{0}\right)}}}{\left(-\frac{{{x}{y}}}{{{x}^{{2}}+{y}^{{2}}}}\right)}{a}{l}{o}{n}{g}{x}={y}:-\frac{{1}}{{2}}\)

\(\displaystyle\therefore\lim_{{{\left({x},{y}\right)}\rightarrow{\left({0},{0}\right)}}}{\left(-\frac{{{x}{y}}}{{{x}^{{2}}+{y}^{{2}}}}\right)}=\div{e}{r}\ge{s}\)

Similary,

\(\displaystyle\Rightarrow\lim_{{{\left({x},{y}\right)}\rightarrow{\left({0},{0}\right)}}}{\left({y}{\ln{{\left({x}^{{2}}+{y}^{{2}}\right)}}}\right)}=\)

Convert to polar coordinates \(\displaystyle\rightarrow{x}={r}{\cos{{\left({0}\right)}}},{y}={r}{\sin{{\left({0}\right)}}}\)

\(\displaystyle=\lim_{{{r}\rightarrow{0}}}{\left({r}{\sin{{\left({0}\right)}}}{\ln{{\left({\left({r}{{\cos{{\left({0}\right)}}}^{{2}}+}{\left({r}{\sin{{\left({0}\right)}}}^{{2}}\right)}\right)}\right.}}}\right.}\)

if \(\displaystyle\lim_{{{x}\rightarrow{a}^{{-}}}}{f{{\left({x}\right)}}}=\lim_{{{x}\rightarrow{a}^{+}}}{f{{\left({x}\right)}}}={L}{t}{h}{e}{n}\lim_{{{x}\rightarrow{a}}}{f{{\left({x}\right)}}}={L}\)

\(\displaystyle\lim_{{{r}\rightarrow{0}^{{-}}}}{\left({r}{\sin{{\left({0}\right)}}}{\ln{{\left({\left({r}{{\cos{{\left({0}\right)}}}^{{2}}+}{\left({r}{\sin{{\left({0}\right)}}}^{{2}}\right)}\right)}={0}\right.}}}\right.}\)

\(\displaystyle\lim_{{{r}\rightarrow{0}^{+}}}{\left({r}{\sin{{\left({0}\right)}}}{\ln{{\left({\left({r}{{\cos{{\left({0}\right)}}}^{{2}}+}{\left({r}{\sin{{\left({0}\right)}}}^{{2}}\right)}\right)}={0}\right.}}}\right.}\)

\(\displaystyle\therefore\lim_{{{\left({x},{y}\right)}\rightarrow{\left({0},{0}\right)}}}{\left({y}{\ln{{\left({x}^{{2}}+{y}^{{2}}\right)}}}\right)}={0}\)

Find two dofferent paths to approach the point that gives different values for

Limit of \(\displaystyle\lim_{{{\left({x},{y}\right)}\rightarrow{\left({0},{0}\right)}}}{\left(-\frac{{{x}{y}}}{{{x}^{{2}}+{y}^{{2}}}}\right)}{a}{l}{o}{n}{g}{x}={0}:{0}\)

Limit of \(\displaystyle\lim_{{{\left({x},{y}\right)}\rightarrow{\left({0},{0}\right)}}}{\left(-\frac{{{x}{y}}}{{{x}^{{2}}+{y}^{{2}}}}\right)}{a}{l}{o}{n}{g}{x}={y}:-\frac{{1}}{{2}}\)

\(\displaystyle\therefore\lim_{{{\left({x},{y}\right)}\rightarrow{\left({0},{0}\right)}}}{\left(-\frac{{{x}{y}}}{{{x}^{{2}}+{y}^{{2}}}}\right)}=\div{e}{r}\ge{s}\)

Similary,

\(\displaystyle\Rightarrow\lim_{{{\left({x},{y}\right)}\rightarrow{\left({0},{0}\right)}}}{\left({y}{\ln{{\left({x}^{{2}}+{y}^{{2}}\right)}}}\right)}=\)

Convert to polar coordinates \(\displaystyle\rightarrow{x}={r}{\cos{{\left({0}\right)}}},{y}={r}{\sin{{\left({0}\right)}}}\)

\(\displaystyle=\lim_{{{r}\rightarrow{0}}}{\left({r}{\sin{{\left({0}\right)}}}{\ln{{\left({\left({r}{{\cos{{\left({0}\right)}}}^{{2}}+}{\left({r}{\sin{{\left({0}\right)}}}^{{2}}\right)}\right)}\right.}}}\right.}\)

if \(\displaystyle\lim_{{{x}\rightarrow{a}^{{-}}}}{f{{\left({x}\right)}}}=\lim_{{{x}\rightarrow{a}^{+}}}{f{{\left({x}\right)}}}={L}{t}{h}{e}{n}\lim_{{{x}\rightarrow{a}}}{f{{\left({x}\right)}}}={L}\)

\(\displaystyle\lim_{{{r}\rightarrow{0}^{{-}}}}{\left({r}{\sin{{\left({0}\right)}}}{\ln{{\left({\left({r}{{\cos{{\left({0}\right)}}}^{{2}}+}{\left({r}{\sin{{\left({0}\right)}}}^{{2}}\right)}\right)}={0}\right.}}}\right.}\)

\(\displaystyle\lim_{{{r}\rightarrow{0}^{+}}}{\left({r}{\sin{{\left({0}\right)}}}{\ln{{\left({\left({r}{{\cos{{\left({0}\right)}}}^{{2}}+}{\left({r}{\sin{{\left({0}\right)}}}^{{2}}\right)}\right)}={0}\right.}}}\right.}\)

\(\displaystyle\therefore\lim_{{{\left({x},{y}\right)}\rightarrow{\left({0},{0}\right)}}}{\left({y}{\ln{{\left({x}^{{2}}+{y}^{{2}}\right)}}}\right)}={0}\)