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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) ?What does it mean for a function ƒ(x, y) to have limit L as (x, y) S (x0 , y0)?Find the limit (if it exists). If the limit does not exist, explain why. lim (xy+yz^2+xz^2/x^2+y^2+z^2) (x,y,z)--->(0,0,0)
- Find the solution of the limit using polar coordinate system. Lim (x,y)->(0,0)√x^2+y^2/x^2+y^2What does it mean for a function ƒ(x, y) to have limit L as (x, y) → (x0 , y0)? What are the basic properties of limits of functions of two independent variables?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.Show limit (x,y) -> (0,0) f(x,y) is invalidCompute 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.Show that the function f (x, y)=(2 + x-y) / [1+ 2x ^ 2)+(3y ^ 2)] ∈R has a limit at (0,0).Calculate the value of the limit lim(x,y)→(0,0) 22xy/x^2+y^2 as we approach (0,0) along the path γ(t)=(−t,t).