## University Calculus: Early Transcendentals (3rd Edition)

Consider $f(x,y)=\dfrac{x^4}{x^4+y^2}$ Let us consider the first approach: $(x,y) \to (0,0)$ along $y=0$ Then, we get $\lim\limits_{x \to 0}\dfrac{x^4}{x^4+(0)^2}=1$ Now, consider the second approach: $(x,y) \to (0,0)$ along $y=x^2$ Then, we get $\lim\limits_{x \to 0}\dfrac{x^4}{x^4+(x^2)^2}=\dfrac{1}{2}$ This shows that there are different limit values for different approaches and this implies that the limit does not exist for the given function.