Question

Find the limits, if they exist lim(x,y)→(0,0) (3x^2/(4x^2+4y^2))

Answer 1

(1) Along the x-axis, we have y = 0, so that

`\displaystyle\lim_((x,y)\to(0,0)) (3x^2)/(4x^2+4y^2) = \lim_(x\to0) (3x^2)/(4x^2) =\lim_(x\to0)\frac34= \frac34`

(2) Along the y-axis, we take x = 0, then

`\displaystyle\lim_((x,y)\to(0,0)) (3x^2)/(4x^2+4y^2) = \lim_(y\to0) \frac0{4y^2} = 0`

(3) Along the line y = mx, we have

`\displaystyle\lim_((x,y)\to(0,0)) (3x^2)/(4x^2+4y^2) = \lim_(x\to0) (3x^2)/(4x^2+4(mx)^2) = \lim_(x\to0)(3x^2)/(4x^2+4m^2x^2) = \frac3{4+4m^2}`

which means the limit is dependent on the slope of the line m.

(4) nonexistent