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alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

Iv value does not exist but limit does let fx x2 a2

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Unformatted text preview: be the Limit f(x) . x→a Case b ≠ 0 : choose ε = C 1 -- b C choose 2 No matter how close x is to a , that is to say no matter how small the δ w e choose, there will always be some r a tional n umbers such that 0 < C x− aC < δ . If x is rational then C f(x) --bC = C 1 -- bC > ε = C 1 -- b C . 2 So b ≠ 0 cannot be the Limit f(x) . → x a Since a can be any point on the real number line, Limit f(x) does not exist anywhere. x→a 36 10. ALUE V AL UE v er sus LIMIT So far when we took Limits of functions as x → a or δ x → 0 we ASSUMED that these Limits exist. But this is not so simple. 1. Does the Limit exist ? 2. Is the Limit from the right = Limit from the left ? i.e. is L imit f(x) = L imit f(x) -x→a x>a right 3. Is the x → a+ x<a left Value f(x) = L imit x→a x=a f(x) In finding the Limit w e first simplify and then put x = a or x = 0 or whatever and evaluate. We can simplify because: δ x only TENDS TO 0, δx ≠ 0 , x only TENDS TO a, x ≠ a , x only TENDS TO 0, x ≠ 0 . Whereas in finding the Value w e must directly put x = a or x = 0 or whatever and evaluate. 37 Let us compare Value f(x) and L imit f(x) x=a We have the following possibilities: x→a f(x) = x (i) Value and Limit both exist and are equal : (ii) Value and Limit exist but are not equal. Let f(x) = { L imit x → a- 2 for x = a 1 for x ≠ a imit f(x) = 1 = L→ + f(x) , Value f(x) = 2 x a x=a Try drawing the graph of this function. (iii) Value exists but Limit does not. Let f(x) = 1 -- x for x ≤ 1 { 3 -- x for 1 < x L imit f(x) = 0 ≠ L imit f (x) = 2 , Value f(x) = 0 x → 1+ x=1 x → 1- Try drawing the graph of this function. (iv) Value does not exist but Limit does. Let f(x) = x2 -- a2 , L imit x -- a x→a f(x) = 2a, Value x=a f(x) = ? L Let f(x) = sin x , x imit f(x) = 1, Value f(x) = ? →0 x x=0 (v) Value and Limit do not exist. 1 Let f(x) = -x , L imit f (x) = ? , Value x→0 x=0 38 f(x) = ? 1 1. CONTINUITY of a Function Our common perception of continuous is so taken for granted that we se...
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