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Problem of the Day

Find the domain of f(x) = 1. 3-2x. Problem of the Day. Find the domain of f(x) = 1. 3-2x. ( - , 3 ). 2. Problem of the Day. Properties of Limits. Direct Substitution. Let b and c be real numbers and n a positive integer -

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Problem of the Day

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  1. Find the domain of f(x) = 1 3-2x Problem of the Day

  2. Find the domain of f(x) = 1 3-2x ( - , 3 ) 2 Problem of the Day

  3. Properties of Limits Direct Substitution Let b and c be real numbers and n a positive integer - a) lim b = b (lim 4 = 4) b) lim x = c (lim x = 5) c) lim x = c (lim x = 9) x c x 5 x c x 5 2 2 2 x c x 3

  4. x c x c x c x c Properties of Limits Let b and c be real numbers and n a positive integer and f and g functions with limits L and K respectively - scalar multiple lim [b f(x)] = bL sum or difference lim [f(x) +g(x)] = L + K product lim [f(x)g(x)] = LK quotient lim f(x) = L power lim [f(x)] = L g(x) K n n x c

  5. lim (4x + 3) = x 2 Properties of Limits 2

  6. 2 lim 4x + lim 3 x 2 x 2 2 4 lim x + lim 3 x 2 x 2 2 4(2 ) + 3 Properties of Limits 2 lim (4x + 3) = x 2 19

  7. x c Properties of Limits If p is a polynomial function and c is a real number then lim p(x) = p(c). If r is a rational function r(x) = p(x), c is a real number and q(c) = 0 then q(x) lim r(x) = r(c) = p(c) q(c) x c

  8. Properties of Limits 2 lim x + x + 2 x 1 x + 1

  9. Properties of Limits 2 2 lim x + x + 2 1 + 1 + 2 = x 1 x + 1 1 + 1 = 2

  10. n n lim n = c x c Properties of Limits Let n be a positive integer. If n is odd and c is a real number or if n is even and c is > 0 then If f and g are functions with limits L and M then lim f(g(x)) = f(L) x c

  11. x c x c x c x c x c x c Properties of Limits Trig Functions lim sin x = sin c lim cos x = cos c lim tan x = tan c lim cot x = cot c lim sec x = sec c lim csc x = csc c

  12. lim (x cos x) = x Properties of Limits

  13. Properties of Limits lim (x cos x) = x (lim x)(lim cos x) x x cos -

  14. ? ? ? How observant have you been? Can you draw the figure that is a composite of all the parts that you have been shown in the upper right hand corner of the screens? ? ? ? ? ? ?

  15. Strategy for Finding Limits x c x c 2 (x - 1)(x + x + 1) 3 lim x - 1 = x 1 x - 1 x - 1 1. Cancellation and direct substitution If f(x) = g(x) for all x = c and c is real (functions that agree at all but one point) Then lim f(x) = lim g(x) 2 = x + x + 1 = 3

  16. Strategy for Finding Limits lim x + 1 - 1 x 0 x x + 1 - 1 ( ) ( x + 1 + 1 ) x x + 1 + 1 2. Rationalization and direct substitution (x + 1) - 1 = 1 x ( x + 1 + 1) 2

  17. x c x c x c Strategy for Finding Limits g(x) 3. Squeeze Theorem f(x) h(x) If h(x) < f(x) < g(x) for all x in the open interval containing c, except possibly c, and lim h(x) = L = lim g(x) then lim f(x) exists and is L.

  18. Strategy for Finding Limits 4. By definition ) ( lim sin x lim x = 1 = 1 x x 0 x 0 sin x lim 1 - cos x = 0 x x 0

  19. Strategy for Finding Limits 5. Be creative! lim sin 4x x x 0 (as x approaches 0, y also approaches 0) (let 4x = y) 4 lim sin 4x 4 lim sin y = = 4(1) x 4 y x 0 y 0

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