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alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

# Sin height yt over view ov er vie w let us continue

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Unformatted text preview: δx → 0 δx → 0 δx → 0 Limit from the RIGHT L imit { f(x) } = L imit x → 0+ x → 0+ { 1 x } = +1 0 = +∞ , something undefined. / / Limit { f(x) } = Limit {f(0 + δ x)} = Limit { 1 (0 + δ x) } =+1 0 = +∞ / / δx → 0 δx → 0 δx → 0 The LIMIT from the left does not exist. And, the LIMIT from the right does not exist. left right Also Value { f(x) } = f(0) = 1 0 = ∞ , something undefined. / x=0 So on any one of the three counts, be it LIMIT from the left or LIMIT from the right left right or the VALUE, f(x) is discontinuous at x = 0. Moreover, we cannot remove this discontinuous discontin discontinuity. 50 sin(x) continuous at x = 0 ? continuous 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 7: Is the function f(x) = Limit from the LEFT sin(− δ x) L imit {f(x)} = L imit { sin(0 − δ x) } = L imit { } δx → 0 δx → 0 δx → 0 − δx (0 − δ x) − δx L L } = δ ximit0 {1} = 1 . = δ ximit0 { → → − δx irst take Note how we simplify ffirst and then take the limit. We can simplify first because simplify ir δx only TENDS TO zero. δx ≠ 0. Limit from the RIGHT L imit {f(x)} = L imit { sin(0 + δ x) } = L imit { sin(+ δ x) } δx → 0 δx → 0 δx → 0 + δx (0 + δ x) δx } = L imit {1} = 1 . δx → 0 δx → 0 δx Note how we simplify ffirst and then take the limit. We can simplify first because irst take simplify ir δx only TENDS TO zero. δx ≠ 0. = L imit { Since, L imit -- { f(x) } = 1 = L imit+ { f(x) } , we say L imit x→0 x→0 x→0 sin(x) { } =1. x Value { f(x) } = Value { sin(x) } = 0 , which is something undefined. /0 x x=0 x=0 Since L imit f(x) ≠ Value f(x) we say f(x) is discontinuous at x = 0. discontinuous x→0 x=0 sin(x) for x ≠ 0 We may remove this discontinuity by redefining f(x) to be : f(x) ={ x 1 for x = 0 51 Example 8 : Is the function f(x) defined below continuous anywhere ? continuous f(x) = +1 if x is rational rational { 0 if x is irrational irr ir We saw t...
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