Concept explainers
In Exercises 5–34, estimate the given limit numerically if it exists. [HINT: See Examples 1–3.]
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Applied Calculus
- Find the limits in Exercises 33–35 Are the functions continuous at the point being approached?arrow_forwardFind the limits in Exercises 9–12arrow_forwardIn Exercises 1–4, show that the limit leads to an indeterminate form. Then carry out the two-step procedure: Transform the function alge- braically and evaluate using continuity.arrow_forward
- In Exercises 4–18,(evaluate the limit)or explain why it does not exist. 4. lim (x² – 4x + 1) 5. lim x 3 x + 6 x + 3 X4arrow_forwardIn Exercises 55–72, sketch the graph of the function. Indicate the tran- sition points and asymptotes.arrow_forwardProve the limit statements in Exercises 37–50.arrow_forward
- In Exercises 79–82, find a function that satisfies the given conditions and sketch its graph. (The answers here are not unique. Any function that satisfies the conditions is acceptable. Feel free to use formulas defined in pieces if that will help.) 79. lim f(x) = 0, lim f(x) = ∞, and lim f(x) = ∞ x→too x-2+ 80. lim g(x) = 0, lim g(x) = –∞, and lim g(x) = ∞ x→3- x→3* 81. lim h(x) = -1, lim h(x) = 1, lim h(x) = -1, and x -00 lim h(x) = 1 x→0+ 1, lim k(x) x→l¯ = 00, and lim k(x) x→I* 82. lim k(x) = -00arrow_forwardIn Exercises 75–78, sketch the graph of a function y = f(x) that satis- fies the given conditions. No formulas are required-just label the coordinate axes and sketch an appropriate graph. (The answers are not unique, so your graphs may not be exactly like those in the answer section.) 75. f(0) = 0, f(1) = 2, f(-1) = -2, lim f(x) = -1, and x--00 lim f(x) = 1 76. f(0) = 0, lim f(x) = 0, lim f(x) = 2, and lim f(x) = -2 x→0* %3D 77. f(0) = 0, lim f(x) = 0, lim f(x) = lim f(x) = ∞, x-too x→1- x--1+ = -0, and lim f(x) = -∞ lim f(x) x→1* 78. f(2) = 1, f(-1) = 0, lim f(x) = 0, lim f(x) = ∞, x→0* lim f(x) = -00, and lim f(x) = 1 X -00arrow_forwardThe process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x: Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36.arrow_forward
- Calculate each limit in Exercises 35. lim -4x-3 36. lim 2x-3/4 37. lim 2r-4/3 38. lim -5x3/5 x00 x-00 39. lim (V – x) 40. lim (r – x³) x00 41. lim (-3x5 + 4x + 11) 42. lim (5 – 2x + 3x³) x00 X-00 x² + 8x + 16 x+1 43. lim x-4 (x+ 4)2(x+ 1) x² +1 44. lim x-+2 (x – 2)2 x+4 45. lim x-0 x(x – 1) 46. lim x-4 x2 + 8x + 16 x - 1 47. lim X0 x - X 48. lim X1 x2 - 2x +1 (3x + 1)²(x – 1) 1- 2r2 49. lim 50. lim x-00 (3 – x) (3 + 4x) X00 1-x3arrow_forward3. Prove: lim (6 – 2x) = –2 Give both a preliminary analysis and a formal proof. (See Example 2 in Section 2.4 of the textbook)arrow_forwardCompute the indicated limit. „2 x + 3 lim x00 x - 4arrow_forward
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