00b773…: “As for the silver mountain”

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Consider $F(x,y) = \frac {6xy}{5x^2+4y^2}$ . While computing $\lim _{\vector {x,y}\to \vector {0,0}} F(x,y)$ a student provides the following response:

The student’s work is correct. The student’s work is incorrect.

Evaluate the limit along the path $y=mx$ . $\lim _{x\to 0 }F(x,mx) = \answer {\frac {6 m}{5+4m^2}}$

So we conclude:

The limit exists and is equal to $\frac {6 m}{5+4m^2}$ . The limit does not exist.