Let f(x,y)=-(xy)/(x^2+y^2). Find limit of f(x,y) as (x,y)rarr(0,0)i)Along y axis and ii)along the line y=x.Evaluate Limes lim_((x,y)rarr(0,0))y.log(x^2+y^2),by converting to polar coordinates.

Question
Let \(\displaystyle{f{{\left({x},{y}\right)}}}=-\frac{{{x}{y}}}{{{x}^{{2}}+{y}^{{2}}}}\).
Find limit of \(\displaystyle{f{{\left({x},{y}\right)}}}{a}{s}{\left({x},{y}\right)}\rightarrow{\left({0},{0}\right)}{i}{)}{A}{l}{o}{n}{g}{y}{a}\xi{s}{\quad\text{and}\quad}{i}{i}{)}{a}{l}{o}{n}{g}{t}{h}{e}{l}\in{e}{y}={x}.{E}{v}{a}{l}{u}{a}{t}{e}\Lim{e}{s}\lim_{{{\left({x},{y}\right)}\rightarrow{\left({0},{0}\right)}}}{y}.{\log{{\left({x}^{{2}}+{y}^{{2}}\right)}}}\),by converting to polar coordinates.

Answers (1)

2021-02-03
\(\displaystyle\lim_{{{\left({x},{y}\right)}\rightarrow{\left({0},{0}\right)}}}{\left(-\frac{{{x}{y}}}{{{x}^{{2}}+{y}^{{2}}}}\right)}\)
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 \(\displaystyle\lim_{{{x}\rightarrow{a}^{{-}}}}{f{{\left({x}\right)}}}=\lim_{{{x}\rightarrow{a}^{+}}}{f{{\left({x}\right)}}}={L}{t}{h}{e}{n}\lim_{{{x}\rightarrow{a}}}{f{{\left({x}\right)}}}={L}\)
\(\displaystyle\lim_{{{r}\rightarrow{0}^{{-}}}}{\left({r}{\sin{{\left({0}\right)}}}{\ln{{\left({\left({r}{{\cos{{\left({0}\right)}}}^{{2}}+}{\left({r}{\sin{{\left({0}\right)}}}^{{2}}\right)}\right)}={0}\right.}}}\right.}\)
\(\displaystyle\lim_{{{r}\rightarrow{0}^{+}}}{\left({r}{\sin{{\left({0}\right)}}}{\ln{{\left({\left({r}{{\cos{{\left({0}\right)}}}^{{2}}+}{\left({r}{\sin{{\left({0}\right)}}}^{{2}}\right)}\right)}={0}\right.}}}\right.}\)
\(\displaystyle\therefore\lim_{{{\left({x},{y}\right)}\rightarrow{\left({0},{0}\right)}}}{\left({y}{\ln{{\left({x}^{{2}}+{y}^{{2}}\right)}}}\right)}={0}\)
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