alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

# limit from the left 1 tends to step near 0 and to

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Unformatted text preview: hing undefined. /0 x -- a x=a x=a n n The reader must have guessed by now that L imit { xx -- a } = n a n−1 . -- a → x a deriv tiv We shall see this very special limit in its proper context when we study the derivative deri of f(x) = xn . 29 Example 6 : What is the LIMIT of f(x) = + √x as x TENDS TO 0 ? f(x) = +√ x (0, 0) x Limit from the LEFT 1. TENDS TO step : near 0 and to the left of 0 : left x = 0 − δx f(x) = √ (0 − δx) = √ − δx A negative real number under the radical sign is a complex number. We are dealing with real numbers and real valued functions. Here √ − δx is not real valued and so it is not defined. f(x) = + √x is not defined for x < 0. not left 2. LIMIT step : The LIMIT from the left does not exist. Limit from the RIGHT 1. TENDS TO step : near 0 and to the right of 0 : right x = 0 + δx f(x) = √ (0 +δx) = √δx 2. LIMIT step : L imit δx → 0 {√δx} = 0 Since L imit − f(x) ≠ L imit + f(x) we say : L imit f(x) does NOT exist. x→0 x→0 x→0 Value x = 0 f(x) = f(0) = √0 = 0 30 Example 7: What is the LIMIT of f(x) = 1 x as x TENDS TO 0 ? / Limit from the LEFT 1. TENDS TO step : near 0 and to the left of 0 : left x = 0 − δx f(x) = f(0 − δ x) = { 1 (0 − δ x) } = − 1 δ x . Note that δx ≠ 0 . / / 2. LIMIT step : L imit δx → 0 { − 1 δ x }= − 1 0 = −∞ , something undefined. / / Limit from the RIGHT 1. TENDS TO step : near 0 and to the right of 0 : right x = 0 + δx f(x) = f(0 + δ x) = { 1 (0 + δ x) } =+1 δ x . Note that δx ≠ 0 . / / L 2. LIMIT step : δ ximit0 {+1 δ x }= =+1 0 = +∞ , something undefined. / / → The LIMIT from the left does not exist. And, the LIMIT from the right does not exist. left right 1 = ∞ , something undefined. VALUE { f(x) } = f(0) = / 0 x=0 +∞ / f(x) = 1 x 2 1 ½ −∞ 0 ½1 −1 −2 −∞ 31 2 ... +∞ sin(x) as x TENDS TO 0 ? x We shall use the trigonometric identity sin (A ± B) = sin A cos B ± cos A sin B. Also, when x is very small, we may say* : sin (x) = x . Example 8: What is the LIMIT of f(x) = Limit from the LEFT 1. TENDS TO step...
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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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