Show whether or not the following limit exists: xy³ lim (x,y)→(0,0) [x2 + y6
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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/(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.Let f (x, y) = x3 + y3/xy2 . Set y = mx and show that the resultinglimit depends on m, and therefore the limit lim (x,y)→(0,0) f (x, y) doesnot exist.
- 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.Use two different methods to find the limit where m > 0, n > 0, and x > 0Show that the function f (x, y)=(2 + x-y) / [1+ 2x ^ 2)+(3y ^ 2)] ∈R has a limit at (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.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.1. Use the definition of the limit ( epsolon - delta ) to show thatlim of 1/z as z approaches -i2. Give the condition which ensure that |ez| < 1 where z in C.
