Calculus I Worksheet #8 Review for Test 2 Limits and Continuity

Calculus I Worksheet #8 Review for Test 2 – Limits and Continuity

1 s in 5 x  2 s in 2 3 x  lim lim  2  x 0 c o s 4 x  x 0 x c o s x  3   4  x  s i n 5 x  l i m l i m x  0  1  x  0 t a n x  s i n x   3  5 1  s i n x 6 c o s 2 x l i m lim  2 x  1  c o s x x  x 2 3 7 6 x  5 x 8 4 a 2  x 2  lim 2 3 x  x  4 x lim   x b  2 a  x  9 8 x 3  5 x 10 x 2  x 4 lim 2 lim 2 6 x  x  3 x x  x  x 11 4 x 3  3 2 12 5 l im lim 2  x 2 5 x  20 x  4 x  4 13 4 骣x 14 Find the domain and range of the function lim sin 琪 x 0 x 桫5 f( x )= x2 - 9 + 1 15 4x3 - 5 16 Determine if the following function is even, odd or lim 3 x 1 5x2 - 6 neither: f( x )= 3 x - 5 x 17 2x - 2 18 Write the equation of the line perpendicular to lim x 2+ x - 4 3x - 4y = 12 that passes through the point (-3, 1)

19 8x2- 2 x 3 20 x2 - 4; x -3 lim 2  x� 2x+ 4 x Graph f( x )= x+3; -3 < x < 2  -x+2; x 2 Use the following graph of f(x) to answer the next five questions

 limf ( x ) limf ( x ) limf ( x ) limf ( x ) limf ( x ) 21 + 22 - 23 24 25  x 1 x 1 x 1 x� 1 x 2 

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Use the graph of f(x) below to answer 26 a-c y

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 26a) Use the 3-part definition of continuity to show if f(x) is continuous at x = 3 26b) What kind(s) of discontinuity are shown in the graph of f(x)? 26c) If there is a removable discontinuity? If so, assign a value to remove it. 27 2x - 1; x 1 If f( x ) = use the definition of continuity to show if f(x) is continuous at x = 1 -3x+1; x>1 28 x2 +6 x + 8 ; x -2 If f( x ) = x + 2 use the definition of continuity to show if f(x) is continuous at x = -2 2; x=-2 29 x3 + 8 If f( x ) = use the definition of continuity to show if f(x) is continuous at x = -2 x + 2 30 x3 - 64 When f( x ) = , then f(x) has point discontinuity. Assign a value to f(x) that removes the x - 4 discontinuity.

Answers: 3 1. 0 2. 9 3. 15 4. 1 5. 2 6. 0 7. 2 - 12 - 4 8. 2a-b 9. 10. 0 11. 12. 13. 14. 5 5 d= { x� 3 or x 3} r = [1, ) 15. 1 16. odd 17. –1 -4 19. 20. graph 21. 1 18. y-1 =( x + 3) 3 below 22. –2 23. DNE 24. 2 25. 1 26a. 26b. point, 26c. yes let f(x) =2 below & infinite 27. 28. 29. 30. let f(4) = 48 below below below see below

y  1. f (3) exists f(3) = 3  [ ]

  轾  2.limf ( x ) exists lim f ( x )= lim f ( x ) = 2 26a. + -  x 3 臌犏x 3 x 3   3.f (3)筡 lim f ( x ) f ( x ) is not continuous at x=3  x 3 

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20.  1. f (- 2) exists [ f(-2) = 2] 27. 2. limf ( x ) exists 轾 lim f ( x )= lim f ( x ) = 2 28. + - 1. f (1) exists [ f(1) = 1] x� 2 臌犏x��2 x 2 3.f (- 2) = lim f ( x ) = 2 \ f ( x ) is continuous at x=-2 2.limf ( x ) DNE 轾 lim f ( x )= 1� lim f ( x ) - 2 x� 2 x 1 臌犏x1+ x 1 - \ f( x ) is not continuous at x = 1 2 1. f (- 2) DNE [ f(-2) = 12] x3 - 64 ( x-4)( x + 4 x + 16) 29. 30.  ( x2 +4 x + 16) if x 4 \ f( x ) is not continuous at x=-2 x-4( x - 4)