ASSESSMENT - Limits & Continuity

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The exam I use to assess Calculus Unit 2 - Limits & Continuity. It is primarily made of AP level questions, both mul...

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UNIT 2 – ASSESSMENT – REVISED 1.

NAME:_____________________________________________

2.

3.

4.

5. If

lim xa

a) b) c) d)

f ( x)  L ,

where L is a real number, which of the following must be true?

f(x) is continuous at x=a f(x) is defined at x=a f(a)=L none of these

6. If f is continuous on  4,4 such that f(-4) =11 and f(4) = -11, then a) f(0) =0 b) lim f ( x)  8 x 2

c) there is at least one c   4,4 such that f(c)=8 d)

lim f ( x)  lim f ( x) x 3

x 3

e) It is possible that “f” is not defined at x=o.

7.

R x  25 | 8. h( x )  Sx  5 ||10 T 2

x5

Which of the following statements I, II, and III are TRUE? x5

I) lim h( x) exists x5

II) h(5) exists III) h is continuous at x = 5 A) Only I

B) Only II

C) I and II

D) II and III

R || xx 11 |3 f ( x)  S || |Tx  3 2

9. _____ 9. Let f be a function defined by

A) 0

10. If xlim  6

B) 2

E) I, II and III

x 1 x 1

Find lim f ( x)  x1

x 1

C) 3

D) 4

E) does not exist

C) 0

D) 2

E) 

2 x6

A) 

x6

= B) -2

11. Let

2 x  4  g ( x )   cx  d  5 x2 

x3 if 3 x1 if x1 if

Find the value(s) of c and d that make g continuous at all real values of x.

12. Use the Intermediate Value Theorem to show there is a solution to f ( x)  x3  8x2  2 x  5  18 . Find a oneunit interval.

13. Find the points, if any, at which the functions are not continuous. State the type of discontinuity. Show work that leads to your answer.  2 x2  x  6 if   x2  3x  6 if 

x2 x2

14. Sketch the graph of a function that satisfies all of the following conditions.

lim f ( x)  2,

x 0

lim f ( x)  1,

x 0

y

f (0)  1

 

lim f ( x)  ,

x2



lim f ( x)  



x2



lim f ( x)  4,

x 3









 

f (3)  2

   

Find the limits

 x 3  125 ,  lim x  5  15. x 5  2, 

1 3x lim x  1 3x  5x

x 5 16.

x=5

x2  4 x  8 lim x  3x3

x

17.

18. xlim 3

x 2  3x x2  6 x  9







x 

2 2 19. Given that 4  x  f ( x)  4  x for all x close to x=0. Find

20. Determine whether the following limits exist. If they do, then find the limit. If not, write DNE. a.

lim f ( x)

b.

lim d ( x)

c.

lim f ( x)

x 3

x 2

x 1

d ( x) d. xlim  4 e.

f ( x) x  0 d ( x)

lim

f ( x)d ( x) f. lim x 2 g. lim x 0

d ( x)  3 f ( x)

h.

lim (2 f ( x)  d ( x))

x  2

lim f ( x) x 0

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