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Using the subscript notation for partial derivatives, we have V Vx x Vy y Vx cos Vy sin V Vx x Vy y Vx sin Vy cos Dividing both sides of (2) by , we have 1 V Vx sin Vy cos Then from (1) and (3), we have

FullSimplify[messyexpression]

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5 (11 + 3 x(4+ x)) 5 5 5 (1 + x) (2 + x) (3 + x)

PARTIAL DERIVATIVES @z @z @u f 0 u 2xy; @x @u @x Then x @z f 0 u 2x2 y; @x 2y @z @z @u f 0 u x2 @y @u @y

messytrigexpression = Expand[(Tan[x]2 + Sin[x]2 + Cos[x]2)5] Cos[x]10 + 5 Cos[x]6 Sin[x]2 + 5 Cos[x]8 Sin[x]2 + 10 Cos[x]2 Sin[x]4 + 20 Cos[x]4 Sin[x]4 + 10 Cos[x]6 Sin[x]4 + 30 Sin[x]6 + 30 Cos[x]2 Sin[x]6 + 10 Cos[x]4 Sin[x]6 + 20 Sin[x]8 + 5 Cos[x]2 Sin[x]8 + Sin[x]10 + 10 Sin[x]4 Tan[x]2 + 30 Sin[x]6 Tan[x]2 + 5 Sin[x]8 Tan[x]2 + 20 Sin[x]4 Tan[x]4 + 10 Sin[x]6 Tan[x]4 + 5 Sin[x]2 Tan[x]6 + 10 Sin[x]4 Tan[x]6 + 5 Sin[x]2 Tan[x]8 + Tan[x]10 Simplify[messytrigexpression] Sec[x]10

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The limit of a function is the foundation stone of differential calculus. For a complicated function, the calculation of a limit can be quite difficult and can require specialized techniques for its evaluation. Mathematica has built-in procedures for accomplishing this task and always attempts to determine the exact value of the limit.

Limit[f[x], x a] computes the value of lim f ( x ).

Summary

Another method: We have Also, Then dz f 0 x2 y d x2 y f 0 x2 y 2xy dx x2 dy : dz @z @z dx dy: @x @y @z x3 f 0 x2 y . @y x @z @z 2y . @x @y

5 EXAMPLE 1 We wish to compute lim x 3 32 . Because both numerator and denominator approach zero as x 2,

x 2

x 8

6.25. If for all values of the parameter and for some constant p, F x; y p F x; y identically, where F is assumed di erentiable, prove that x @F=@x y @F=@y pF.

Direction 1 causes the limit to be computed as a left-hand limit with values of x approaching a from below. Direction 1 causes the limit to be computed as a right-hand limit with values of x approaching a from above.

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The default for the Limit command is Direction Automatic, which provides Direction 1 except for limits at . Thus, Mathematica may give a misleading representation of the limit of a discontinuous function if the Direction option is omitted.

x 0

Let x u, y v. Then F u; v p F x; y The derivative with respect to of the left side of (1) is @F @F @u @F dv @F @F x y @ @u @ @v @ @u @v The derivative with respect to of the right side of (1) is p p 1 F. @F @F y p p 1 F x @u @v Letting 1 in (2), so that u x; v y, we have x @F=@x y @F=@y pF. Then 2 1

Limit Abs[x], x 0 x 1 By default, only the right-hand limit has been computed, since no direction was specified. To fully analyze the limit we must compute the left-hand limit as well. Limit Abs[x], x 0, Direction 1 x 1 The limit does not exist since the left- and right-hand limits are different numbers.

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Limit[1/x, x 0, Direction 1] Limit[1/x, x 0, Direction 1]

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