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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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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.
 Fall '09
 TAMERDOğAN

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