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alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

# limit xa x a irst take simplify ffir st then tak e

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Unformatted text preview: to know the behaviour of a function near any chosen instant a : a little to the left of a or (a − δx) and a little to the right of a or (a + δx). left right 39 Geometrically Geometrically when we say that a SINGLE VALUED function is CONTINUOUS around the point or instant a we mean that the curve of the function is without a, breaks or gaps. f(x) 3 value f(a)=3 f(x) Limit +f(x) = 3 x→a 2 1 (0,0 0,0) ( 0,0 ) Limit - f(x) =3 x→a a -- δ x a a +δ x x f(x) is CONTINUOUS at the point or instant a f(x) 3.5 3 Limit - f(x) =3 x→a value f(a)=3 Limit +f(x) = 3.5 x→a 2 1 (0 , 0) a -- δ x a a +δ x x f(x) is CONTINUOUS at instant a bu t only from the left l eft f(x) value f(a)=3.5 Limit - f(x) =3 x→a 3.5 3 Limit +f(x) = 3.5 x→a 2 1 0,0) (0, 0) a -- δ x a a +δ x x f(x) is CONTINUOUS at instant a bu t only from the right r ight 40 Analytically : Assume Value f(x) = f(a) exists and is something definite. Then x=a 1. if xL imit -- f(x) exists and is = Value f(x) →a xa = we say that f(x) is CONTINUOUS at the point x = a from the left . l eft 2. if L imit+ f(x) exists and is = Value f(x) x→a x=a we say that f(x) is CONTINUOUS at the point x = a from the right . r ight f(x) is said to be CONTINUOUS at the point or instant x = a point instant if L imit f(x) = f(a) = L imit f(x) x → a -- x → a+ Using the infinitesimal δ x we can write this as: f(x) is said to be CONTINUOUS at the point or instant x = a if Limit f(a -- δ x) = f(a) = Limit f(a + δ x) δx → 0 δx → 0 If f(x) is not CONTINUOUS at a we say f(x) is DISCONTINUOUS at a. We may test the continuity of a function using the decision tree below. continuity contin f(x) CONTINUOUS at a ? LIMIT x→a ? = ALUE VALUE f(x) = f(a) x=a directly evaluate with x = a LIMIT -LIMIT ? = LIMIT x→a x → a+ irst take simplify ffir st then tak e the limit ir 41 polynomials Are the pol ynomials C ONTINUOUS ever ywhere ? p ol We can easily see that f(x) = xn for n = 1, 2, 3, . . . Limit f(x) = Value f(x) = an x→a x =a Since a can be any point on the real number line, these f(x) are continuous continuous everywhere. Likewise, f(x) = an xn for n = 0, 1, 2, 3, . . . continuous where the coefficients an are real numbers, are also continuous everywhere. contin Combining both we get the general polynomial f(x) : f(x) = a n x n + a n-- 1 x n--1 + . . . + a 2 x 2 + a 1 x + a 0 These are single valued and continuous everywhere. single valued continuous sing contin y(t) = u . s i n θ . t -- 1 g t 2 is a polynom...
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