Limits at Infinity. Use algebraic techniques to help with indeterminate forms of ± Use substitutions to evaluate limits of compositions of functions.

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1 SUGGESTED REFERENCE MATERIAL: Limits at Infinity As you work through the problems listed below, you should reference Chapter. of the recommended textbook (or the equivalent chapter in your alternative textbook/online resource) and your lecture notes. EXPECTED SKILLS: Be able to determine limits at infinity - especially for polynomials, rational functions, functions involving radicals, exponential functions, and logarithmic functions. Use algebraic techniques to help with indeterminate forms of ± ± Use substitutions to evaluate limits of compositions of functions. PRACTICE PROBLEMS: and.. Based on the graph of F (x) shown below, compute the indicated limits. (Make reasonable assumptions about the behavior of the function outside of the shown region.) (a) (b) lim F (x) x lim F (x)

2 . Based on the graph of G(x) shown below, compute the indicated limits. (Make reasonable assumptions about the behavior of the function outside of the shown region.) (a) (b) lim G(x) x lim G(x) For problems -, compute the limit. If the limit doesn t exist write,, or DNE (whichever is most appropriate).. lim 4. lim ( x + ) 5. lim x (x 7)(5 x) 6. lim (5 x + 4x ) 5

3 ( lim x 00 ) 0 ( ) x + 8. lim x ( ) x + 9. lim x 0 ( ) x 0. lim x (. lim. lim x 5 4x x 5 4x 4x + 5 ( ) x 4 x x 0 ( ) x 4x + 6. lim 9 x ( ) x x + 4. lim x + 0 ( ) 4 + x 5. lim + x )

4 6. lim ( ) 4 + x + x ( ) x 5x lim 8 9x x ; Video Solution: 8. lim e x 9. lim ex 0 ( ) 0. lim e x. lim e /x ( ) 7. lim e x 8 0 ( ) 7. lim e x lim x 0 + ln x 5. lim ln x 4

5 ( ) ln 6x 6. lim ln x 7. lim [ln (x + ) ln (x + 5)] ( ) ln ( ) 8. lim x + 8x 5 x 4 ( 9. lim x + ) x + x ( 0. lim x + ) x + x ( ). lim x x x. A tank contains 5000 liters of pure water. Brine containing 0 grams of salt per liter of water is pumped into the tank at a rate of 5 liters per minute. It can be shown thta the concentration of salt in the tank after t minutes is: What happens as t? C(t) = The concentration of salt approaches 0 g/l 0t 00 + t Use the following two definitions to answer questions -6. Definition: A function f(x) has a horizontal asymptote of y = L if at least one of the following is true: lim f(x) = L lim f(x) = L. 5

6 Definition: A function f(x) has a vertical asymptote of x = a if at least one of the following is true: lim f(x) = x a lim f(x) = x a lim f(x) = x a + lim f(x) = x a +. Compute the equations of all horizontal asymptotes and vertical asymptotes, if any, for each of the following functions. (a) f(x) = 4x x Vertical Asymptote: x =, Horizontal Asymptote: y = 4 (b) f(x) = x 5x + 4 x 6x + 8 Vertical Asymptote: x =, Horizontal Asymptote: y = 4. Let y = f(x) satisfy the following: lim f(x) = lim f(x) = 7 lim f(x) = x 6 + lim f(x) = x 6 Based on this information, determine equations for the horizontal and vertical asymptotes of f(x). Vertical Asymptote: x = 6; Horizontal Asymptote: y = 7 5. Sketch a function y = f(x) which satisfies the following conditions. (There are many possible answers.) f() = 0 lim f(x) = lim f(x) = 0 lim x f(x) = lim f(x) = x + 6

7 lim x 0 f(x) = 6. Determine whether the following statement is true or false. If the statement is true, explain why. If the statement is false, provide a specific counterexample. A function y = f(x) can have at most two horizontal asymptotes. True. We determine the horizontal asymptotes of y = f(x) by computing the end behavior; i.e., we compute f(x) and lim f(x). Having a finite value for either lim x of these limits will yield a horizontal asymptote. So, if lim x f(x) = L and lim f(x) = M (where L and M are distinct, finite, real numbers), then f(x) has two horizontal asymptotes y = L and y = M. For a specific example of a function with two horizontal asymptotes, consider f(x) = tan (x). 7. Consider f(x) = x +. (a) Estimate the area between the graph of f(x) and the x-axis on the interval [0, 6] using rectangles of equal width and right endpoints, as in the diagram below. Is your estimate an overestimate or an underestimate of the actual area? 7

8 A 4; This is an overestimate. (b) Estimate the area between the graph of f(x) and the x-axis on the interval [0, 6] using rectangles of equal width and right endpoints, as in the diagram below. Is your estimate an overestimate or an underestimate of the actual area? How does this estimate compare to your estimate from part (a)? A 8; This is an overestimate; but, it is closer to the actual area than the estmate from part (a). (c) It can be shown that an estimate of the area between the graph of f(x) and the x-axis on the interval [0, 6] using n rectangles of equal width and right endpoints 6(n + )(n + ) can be expressed as A(n) = + 6. Compute lim A(n) and n n interpret your answer. Depicted below are n = 0 rectangles lim A(n) = 78. This is the exact area between the graph of f(x) and the x-axis n on [0, 6] 8