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EXAMPLE 31 First let us generate a messy algebraic expression.
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5 messyexpression = Expand 1 + 1 + 1 x +1 x + 2 x + 3 1 1 5 10 10 + + + + 5 5 4 2 3 3 2 + (1+ x) (2+ x) (1+ x)(2+ x) (1+ x) (2+ x) (1+ x) (2+ x)
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5 10 5 1 5 + 4 5 + 4 + 4 + 2 3 + (1+ x) (2+ x) (3+ x) (1+ x)(3+ x) (2+ x)(3+ x) (1+ x) (3+ x) 10 20 10 10 2 3 + 3 + 3 2 + 3 2 + (2+ x) (3+ x) (1+ x)(2+ x)(3+ x) (1+ x) (3+ x) (2+ x) (3+ x) 5 5 30 30 + + 2 2 + 2 2 + 4 4 (1+ x)(2+ x) (3+ x) (1+ x) (2+ x)(3+ x) (1+ x) (3+ x) (2+ x) (3+ x) 20 30 20 + + 3 2 2 3 (1+ x)(2+ x) (3+ x) (1+ x) (2+ x) (3+ x) (1+ x) (2+ x)(3+ x)
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Now we will simplify. Of course, Mathematica does not remember how messyexpression was generated. Simplify[messyexpression]
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5 (11 + 12 x + 3 x2) 2 5 (6 + 11 x + 6 x + x3)
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FullSimplify[messyexpression]
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5 (11 + 3 x(4+ x)) 5 5 5 (1 + x) (2 + x) (3 + x)
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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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C HA PTE R 8
Differential Calculus
8.1 Limits
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 ).
x a
5 EXAMPLE 1 We wish to compute lim x 3 32 . Because both numerator and denominator approach zero as x 2,
the limit is not immediately obvious.
5 Limit x 3 32 , x 2 x 8 20 3
x 2
x 8
Left- and right-hand limits can be computed with the Direction option.
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.
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.
EXAMPLE 2 Evaluate lim
x 0
x . x
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.
Differential Calculus
Mathematica can compute infinite limits and limits at .
EXAMPLE 3
Limit[1/x, x 0, Direction 1] Limit[1/x, x 0, Direction 1]
2 Limit 2 x + 3 x + 4 , x x2 + 1
The functi ons in the next example exhibit a different behavior. As x 0, the function oscillates an infinite number of times. Mathematica returns the limit as an Interval object. Interval[{min, max}] represents the range of values between min and max.
EXAMPLE 4
Limit[Sin[1/x], x 0] Interval[{ 1, 1}] Limit[Tan[1/x], x 0] Interval[{ , }]
SOLVED PROBLEMS
8.1 Compute lim
SOLUTION
2x + x 1 . 3x x 0
x Limit 2 + x 1 , x 0 3x
1 (1+ Log[2]) 3
8.2 Compute lim
x 0
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