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

Question

asked Dec 13, 2022 183k views

1 Answer

Dorean · Answer 1 · 2022-12-20T10:24:13+0000

(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