Question

Find the limits, if they exist, or type DNE for any which do not exist.

lim(x,y)→(0,0) (3x^2/(5x^2+4y^2))

1) Along the xx-axis:
2) Along the yy-axis:
3) Along the line y=mxy=mx :
4) The limit is:

Answer 1

we have to evaluate,

1)

we have y = 0 along the x axis hence we can write limit along the x axis is given by,

2)

we have x = 0 along the y axis hence we can write limit along the y axis is given by,

3)

along the line y = mx we have y = mx hence limit along the line y = mx is given by,

It means we can say that limit along the line y = mx depends on value of m

For example,

m = 1 we have,

m = 2 we have,

And so on,

4)

we can see that limit along x axis, limit along y axis and limit along the line y = mx are different values.

As limit along different paths are different we can say that,