Multivariable calculus

Multivariable limits

Whether a limit in two variables exists, and the paths that decide it.

• Limit of xy/(x^2+y^2) at the Origin Does Not ExistThe limit of xy/(x^2+y^2) at (0,0) fails: y = 0 gives 0 while y = x gives 1/2. Full two-path test, the line family, and the polar check.
• Limit of (x^2-y^2)/(x^2+y^2) Does Not Exist at (0,0)Along the x-axis this function equals 1, along the y-axis it equals -1. The limit at the origin does not exist. Worked paths and the polar form.
• Limit of x^2y/(x^4+y^2): Lines Agree, Limit Still FailsEvery straight line into the origin gives 0, but the parabola y = x^2 gives 1/2, so the limit of x^2y/(x^4+y^2) does not exist. Why lines are not enough.
• Limit of xy^2/(x^2+y^4) at the Origin Does Not ExistAll lines give 0 but the sideways parabola x = y^2 gives 1/2, so the limit of xy^2/(x^2+y^4) at (0,0) does not exist. Method and the path-choice trap.
• Limit of x^2/(x^2+y^2) at (0,0) Does Not ExistOn the x-axis this function is 1, on the y-axis it is 0, so the limit of x^2/(x^2+y^2) at the origin does not exist. Paths, polar form, and the trap.
• Limit of (x-y)/(x+y) at the Origin Does Not ExistThe limit of (x-y)/(x+y) at (0,0) does not exist: the x-axis gives 1, the y-axis gives -1, and the function blows up near the line y = -x.
• Limit of sin(xy)/(x^2+y^2) at (0,0) Does Not ExistThe limit of sin(xy)/(x^2+y^2) at the origin fails: y = 0 gives 0, y = x gives 1/2, y = -x gives -1/2. Small-angle method worked out in full.
• Limit of x^2y^2/(x^4+y^4) at (0,0) Does Not ExistEven powers do not save this limit: x^2y^2/(x^4+y^4) gives 0 on the axes and 1/2 on y = x, so the limit at the origin does not exist.
• Limit of x^2y^2/(x^2y^2+(x-y)^2) Does Not ExistAll lines except y = x give 0, but on y = x this function equals 1. The limit at the origin does not exist. Worked paths and why the diagonal is special.
• Limit of (xy+z^2)/(x^2+y^2+z^2) Does Not ExistA three-variable limit that fails: the x-axis gives 0, the line y = x in the plane z = 0 gives 1/2, and the z-axis gives 1. Full path work.
• Limit of (x+y)^2/(x^2+y^2) at (0,0) Does Not ExistThis limit fails: y = -x gives 0, y = 0 gives 1, and y = x gives 2. Expand the square, run the path test, and see the polar form 1 + sin(2 theta).
• Limit of x^3y/(x^6+y^2): Parabolas Are Not EnoughLines and parabolas both give 0, but the cubic y = x^3 gives 1/2, so the limit of x^3y/(x^6+y^2) at the origin does not exist. How to find the right curve.
• Limit of x^2y/(x^2+y^2) at the OriginThe limit of x^2y/(x^2+y^2) at the origin is 0. Three paths agree, then polar coordinates turn that agreement into an actual proof.
• Limit of (x^3-y^3)/(x^2+y^2) at the OriginThe limit of (x^3-y^3)/(x^2+y^2) at the origin is 0. Polar coordinates give a bound of 2r, and the squeeze theorem closes the argument.
• Limit of xy/sqrt(x^2+y^2) at the OriginThe limit of xy/sqrt(x^2+y^2) at the origin is 0. Polar coordinates reduce it to (r/2)sin(2theta), bounded in size by half the distance to the origin.
• Limit of sin(x^2+y^2)/(x^2+y^2) at the OriginThe limit of sin(x^2+y^2)/(x^2+y^2) at the origin is 1. Substituting u = x^2+y^2 turns it into the one variable limit of sin(u)/u.
• Limit of x^2y^2/(x^2+y^2) at the OriginThe limit of x^2y^2/(x^2+y^2) at the origin is 0, by polar coordinates or by the bound |f| <= y^2. Includes why 0/0 is not a verdict.
• Limit of (x^2+y^2)ln(x^2+y^2) at the OriginThe limit of (x^2+y^2)ln(x^2+y^2) at the origin is 0. The shrinking factor beats the diverging logarithm, as u ln(u) tends to 0.
• Limit of (x^4-y^4)/(x^2+y^2) at the OriginThe limit of (x^4-y^4)/(x^2+y^2) at the origin is 0. The denominator divides the numerator exactly, leaving the polynomial x^2 - y^2.
• Limit of arctan(1/(x^2+y^2)) at the OriginThe limit of arctan(1/(x^2+y^2)) at the origin is pi/2. The inside runs to plus infinity from every direction, and arctan has that horizontal asymptote.
• Limit of xyz/(x^2+y^2+z^2) at the OriginThe limit of xyz/(x^2+y^2+z^2) at the origin in three variables is 0, proved by the one line bound |f| <= |z|/2 rather than by testing axes.
• Limit of (x^2+y^2)/(sqrt(x^2+y^2+1)-1) Is 2This limit at the origin is 2. Direct substitution gives 0/0, and multiplying by the conjugate clears it to sqrt(x^2+y^2+1)+1, which is continuous.
• Limit of x^2y/(x^2+y^4) at the OriginThe limit of x^2y/(x^2+y^4) at the origin is 0, since |f| <= |y|. Contrast it with x^2y/(x^4+y^2), which has no limit at all.
• Limit of (x^2+y^2)sin(1/(x^2+y^2)) at the OriginThe limit at the origin is 0 even though the sine factor oscillates forever. The squeeze theorem handles it: the whole function sits between -r^2 and r^2.