By Choquet-Bruhat Y., DeWitt-Morette C., Dillard-Bleick M.

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Sin x 1 Try direct substitution, 0 for x. +sin 2x lim sin xsin x x"0 sin ^ 0h +sin 2 ^ 0h sin ^ 0h 0 =0 + 0 = 00 = 2 Again, you ended up with an indeterminate form. Return to the original limit statement and substitute 2 sinx cosx for sin2x. Then factor, reduce, and use direct substitution. sin x +sin 2x lim sin x x"0 sin x +2 sin x cos x =lim sin x x"0 sin x ^1 +2 cos x h sin x =lim ^1 +2 cos x h x"0 =lim x"0 =1 +2 cos ^ 0h = 1 +2 $ 1 =3 Divide by Largest Power of the Variable When the limit involves a rational function, you can divide all terms by the highest power of the variable in the rational function—or you can multiply by the reciprocal of the highest powered term instead.

Qxp 4/1/08 3:22 PM Page 19 1 chapter An Introduction to Limits 2 Notice that since the left-sided and right-sided limits are different, lim f ^ x h does not exist. x"0 ONE-SIDED LIMITS FOR A POLYNOMIAL FUNCTION 3 2 For the function f ^ x h = x - x - 3x, find lim f ^ x h and lim f ^ x h . 4 6 x " -2 x " -2 - + 1 From the graph at the right, you can f(x) 3 23 see that lim f ^ x h =3 2 and 3 x " -2 2 lim f ^ x h =3 3 . x " -2 - + x -2 3 3 2 f(x)= x6 – x4 – 3x 2 Since the left-sided and right-sided limits are the same, we can say that lim f ^ x h =3 2 .

Qxp 4/1/08 3:32 PM Page 36 Direct Substitution f ^ x h by simply substituting c for x and then evaluating Many times, you can find the limit lim x"c the resulting expression. In this section, you will see examples of how to determine the limits of polynomial, radical, and trigonometric functions, as well as how to determine the limit of a quotient of rational expressions. LIMIT OF A POLYNOMIAL FUNCTION Determine lim ^ x 2 + 3x h x"2 1 Begin with the original limit statement.

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