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alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

# Example 1 consider the function fx c a constant what

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Unformatted text preview: zero in this step. here division by zer ero step − we may substitute (a − δ x) for x in f(x). Likewise, when x → a 2. LIMIT step : here we let δ x = 0 so that x coincides with a. We combine both steps in one expression as: Limit-- {f(x)} or Limit+ {f(x)}. x→a x→a Then we combine both the above into one expression: Limit {f(x)} or L imit {f(x)}. → x Note: More precisely, near = as close as we like . near as close we like 24 a δx → 0 f(x) at instant x = a is f(a). This we call the VALUE of f(x) at x = a, written as : VALUE { f(x) } = f(a) x=a Sometimes we may have : L I M I T { f(x) } = b = VALUE x=a x→a { f(x) } But this is not always the case as we shall see from the examples that follow. Example 1: Consider the function f(x) = C, a constant. What is the LIMIT of f(x) as x TENDS TO a ? Limit from the LEFT L I M I T { f(x) } x → a− = L I M I T { f(a − δ x) } δx → 0 L I M I T { C } = C. x → a− = L I M I T { C } = C. δx → 0 Limit from the RIGHT L I M I T { f(x) } x → a+ = L I M I T { f(a + δ x) } δx → 0 LIMIT { C } = C x → a+ = LIMIT { C } = C δx → 0 Since, L I M I T { f(x) } = C = L I M I T { f(x) } , we say L I M I T { f(x) } = C . x → a+ x→a x → a− VALUE { f(x) } = f(a) = C . x=a Here we have : L I M I T { f(x) } = C = VALUE { f(x) } . x→a x=a 25 Example 2: What is the LIMIT of f(x) = x + a as x TENDS TO a ? Limit from the LEFT L imit { f(x) } = L imit { x + a } = x → a-- x → a-- L imit x → a-- { a + a } = 2a Let us find xL imit -- {f(x)} in 2 steps using δx. →a 1. TENDS TO step : near a and to the left of a : left f(x) = { (a − δ x) + a } = 2a − δ x 2. LIMIT step : L imit {2a − δ x } = 2a x = a − δx δx → 0 Limit from the RIGHT L imit { f(x) } = x → a+ L imit x → a+ {x+a} = L imit x → a+ { a + a } = 2a L Let us find x imit + {f(x)} in 2 steps using δx. →a 1. TENDS TO step : near a and to the right of a : right x = a + δx f(x) = { (a + δx) + a } = 2a + δ x 2. LIMIT step : L imit {2a + δ x} = 2a δx → 0 imit Since, L imit-- { f(x) } = 2a = xL→ a...
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