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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) = 2x 2 + 3y 2/xy. Set y = mx and show that the resulting limit depends on m, and therefore the limit lim (x,y)→(0,0) f (x, y) does not exist.Use the Two-Path test to show that the following limit does not exist. (Use y=x2) lim (x,y)→(0,0) (x2y)/(x2+y)2
- Let ƒ : (a,b) →R be an uniformly continuous function. Finish the proof by showing that the limit lim_(x →b)ƒ(x) exists.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.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) =x2(y−1)x4+(y−1)2. Determine the existence of limit lim(x,y)→(0,1)f(x, y).In this example, we can combine the three simple results following the limit definition with the results in Theorem 1 to calculate the limits. We simply substitute the x- and y-values of the point being approached into the functional expression to find the limiting value.FInd A and B such that g(x) has limits at x= -1 and x=3