alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

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.

Ask a homework question - tutors are online