8. Let f(x, y) be the function x3 + y3 xy? Show that the limit lim(9)→(0,0) f (x, y) does not exist by setting y = mx and f(x, y) : showing that the limit depends on m.
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- If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local extremum offon (a,c) ?Let f(x, y) = xy /(x 2 + y2). Show that f(x, y) approaches zero along the x- and y-axes. Then prove that lim f(x, y) does not exist by (x,y) (0,0)showing that the limit along the line y =xis nonzero.Let f (x, y) = xy/(x2 + y2). Show that f (x, y) approaches zeroalong the x- and y-axes. Then prove that lim (x,y)→(0,0)f (x, y) does not exist by showing that the limit along the line y = x is nonzero.
- Compute the limit (x,y)→(0,0) of 8xy / 2x^2+ 4y^2 along the following paths. (a) Along the y-axis. (b) Along the line y=3x. (c) What can you conclude about the limit?Let f(x,y,z)=x^2+4y^2+2z^2. Does the limit lim_(0,0,0)f(x,y,z) exist? If so, use generalized to 3 dimension to show that this is the limit.How do I prove that the limit does not exist as (x,y)->(0,0) and x cannot equal 0. For (x^2 +4y^2) / x^2.
- Prove that the limit of xy[(x^2 - y^2)/(x^2 + y^2)] is zero as (x, y) goes to (0,0)A function h(x, y) is defined by h(x,y)=(x^2 y)/(〖7x〗^6+y^3 ). Verify the limit over h(x, y) exists at the origin along y = x^2? In either case write also the reason.Use two different methods to find the limit where m > 0, n > 0, and x > 0