 # By considering different path of approach, show that the function f(x,y)=(x^2+y)/y has no limis (x,y) ->(0,0) slaggingV 2021-01-02 Answered
By considering different path of approach, show that the function $f\left(x,y\right)=\frac{{x}^{2}+y}{y}$ has no limis $\left(x,y\right)\to \left(0,0\right)$
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It is known that when finding the limits of the multivariable function we have to find the limits using different paths .
Take the path along the x-axis ,then $\left(x,y\right)\to \left(x,0\right)$
$\underset{\left(x,y\right)\to \left(x,0\right)}{lim}f\left(x\right)lim\left(\left(x,y\right)\to \left(x,0\right)\right)\frac{{x}^{2}+y}{y}=0$
Take the path along the y-axis ,then
$\underset{\left(x,y\right)\to \left(x,0\right)}{lim}f\left(x\right)lim\left(\left(x,y\right)\to \left(x,0\right)\right)\frac{{x}^{2}+y}{y}=1$
Using the different paths , the values of the limits is different , then the limit does not exist.
Thus, the given function $f\left(x,y\right)=\frac{{x}^{2}+y}{y}$ has no limits