alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

X 0 x 0 x0 value fx value x0 x0 sinx x

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Unformatted text preview: : near 0 and to the left of 0 : x = 0 − δx left sin(0 − δ x) − δx sin(− δ x) }={ }=1 } ={ f(x) = { (0 − δ x) − δx − δx Note how we simplify ffirst and then take the limit. We can simplify first because irst take simplify ir δx only TENDS TO zero. δx ≠ 0. There is no division by zero in this step. here division by zer ero step 2. LIMIT step : L imit {1} = 1 δx → 0 f(x) = − 4π − 3π − 2π − 1π 0 +1π sin (x) x +2π +3π +4π . . . x * Note: we may infer this from the basic definition of sin (θ) = opposite side / hypoteneuse. When θ is infinitely small, say δ θ , then sin ( δ θ ) = r δ θ/r = δ θ . Another way is to look at the expansion of sin(x). When x becomes infinitely small, say δ x , then only the first term in the expansion matters. 32 Limit from the RIGHT 1. TENDS TO step : near 0 and to the right of 0 : x = 0 + δx right δx sin(0 + δ x) sin(+ δ x) } =1 }= { }={ f(x) = { δx (0 + δ x) + δx Note how we simplify ffirst and then take the limit. We can simplify first because irst take simplify ir δx only TENDS TO zero. δx ≠ 0. There is no division by zero in this step. here division by zer ero step 2. LIMIT step : L imit δx → 0 {1} = 1 sin(x) imit imit Since, L → -- { f(x) } = 1 = L→ + { f(x) } , we say L imit { x } = 1 . x 0 x 0 x→0 Value { f(x) } = Value x=0 x=0 sin(x) { x } = 0 0 , which is something undefined. / Value Example 9 : We now present an example of a function f(x) where x = a f(x) L imit exists everywhere (i.e. a can be any real number), but the x → a f(x) does NOT exist anywhere. f(x) = +1 if x is rational rational { 0 if x is irrational irr ir Recall what we said about the rationals and irrationals. Between any two rationals there are infinitely many rationals. And between any two rationals there are also infinitely many irrationals. So we can see that as x TENDS TO a, the function f(x) will keep fluctuating. The function f(x) will be either 0 or 1. It will not take on a single specific value. Hence we can say that the Limit f(x) does NOT exist. Since a can x→a be any point on the real number line, the L imit does NOT exist anywhere. On fi...
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This note was uploaded on 11/29/2012 for the course PHYSICS 105 taught by Professor Tamerdoğan during the Fall '09 term at Middle East Technical University.

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