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m180.Limits.practicetest2

m180.Limits.practicetest2 - 2.8 THE DERIVATIVE AS A...

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2.8. THE DERIVATIVE AS A FUNCTION 119 Practice Test 1. Use the given graph of f to state the value of each quantity, if it exists. (a) lim x 1 - f ( x ) (b) lim x 1 + f ( x ) (c) lim x 1 f ( x ) (d) lim x →- 2 f ( x ) (e) f ( - 2) 2. Determine the infinite limit. (a) lim x 5 + x - 7 x - 5 (b) lim x ( - π/ 2) + tan x 3. Evaluate the limit, if it exists. (a) lim x 2 x 2 - 3 x + 2 x - 2 (b) lim x 5 3 x + 1 - 4 5 - x 4. Find the limit if it exists. If it does not exist, explain why. (a) lim x 0 - | x | x
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120 CHAPTER 2. LIMITS AND CONTINUITY (b) lim x 0 + | x | x (c) lim x 0 | x | x 5. (a) State the three part definition for a function f to be continuous at a number a . (b) Sketch the graph of the function f ( x ) = ( x 3 + x 2 x +1 if x 6 = - 1 2 if x = - 1 (c) State where the function from part (b) is discontinuous and state which requirement fails from the definition of continuity. 6. Find the limit. (a) lim x →∞ 2 - 5 x 9 x 2 + 7 (b) lim x →∞ 9 x 2 + 3 x 2 x + 1 7. Suppose that for all x , - x 2 + 4 x - 6 f ( x ) (2 x - 5) Find lim x 1 f ( x ). 8. Sketch a graph of a possible function with the given properties: f (0) = 0 , lim x →±∞ f ( x ) = 0 lim x 0 + f ( x ) = 2 , lim x 0 - f ( x ) = - 2 9. Use the precise definition of limit to prove that lim x 2 ( - 3 x + 1) = - 5. 10. Find the slope of the tangent line to f ( x ) = x 2 - 5 x + 3 at a = 2. Use the formula m tan = lim x a f ( x ) - f ( a ) x - a .
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2.8. THE DERIVATIVE AS A FUNCTION 121 11. Find the equation of the tangent line to f ( x ) = x + 3 at a = 6. 12. Each limit represents the derivative of some function f at some number a . State such an f and a in each case.
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