Problem 1QC: Use Theorem 2.4 to evaluate limx2(2x48x16) and limx1x1x. Problem 2QC Problem 3QC Problem 4QC Problem 1E: How is limxap(x) calculated if p is a polynomial function? Problem 2E: Evaluate limx1(x3+3x23x+1). Problem 3E Problem 4E Problem 5E Problem 6E: Evaluate limx5(4x2100x5). Problem 7E: Applying limit laws Assume limx1f(x)=8, limx1g(x)=3, and limx1h(x)=2. Compute the following limits... Problem 8E: Applying limit laws Assume limx1f(x)=8, limx1g(x)=3, and limx1h(x)=2. Compute the following limits... Problem 9E: Applying limit laws Assume limx1f(x)=8, limx1g(x)=3, and limx1h(x)=2. Compute the following limits... Problem 10E: Applying limit laws Assume limx1f(x)=8, limx1g(x)=3, and limx1h(x)=2. Compute the following limits... Problem 11E: Applying limit laws Assume limx1f(x)=8, limx1g(x)=3, and limx1h(x)=2. Compute the following limits... Problem 12E: Applying limit laws Assume limx1f(x)=8, limx1g(x)=3, and limx1h(x)=2. Compute the following limits... Problem 13E: Assume limx1f(x)=8 limx1g(x)=3, and limx1h(x)=2. Compute the following limits and state the limit... Problem 14E: How are limxap(x) and limxa+p(x) calculated if p is a polynomial function? Problem 15E: Suppose g(x)={2x+1ifx05ifx=0. Compute g(0) and limx0g(x). Problem 16E: Suppose f(x)={4ifx3x+2ifx3. Compute limx3f(x) and limx3+f(x). Problem 17E: Suppose p and q are polynomials. If limx0p(x)q(x)=10 and q(0) = 2, find p(0). Problem 18E: Suppose limx2f(x)=limx2h(x)=5. Find limx2g(x), where f(x) g(x) h(x), for all x. Problem 19E: Limits of linear functions Evaluate the following limits. 11. limx4(3x7) Problem 20E: Limits of linear functions Evaluate the following limits. 12. limx1(2x+5) Problem 21E: Limits of linear functions Evaluate the following limits. 13. limx95x Problem 22E: Limits of linear functions Evaluate the following limits. 15. limx64 Problem 23E: Evaluating limits Evaluate the following limits. 25. limx1(2x33x2+4x+5) Problem 24E: Evaluating limits Evaluate the following limits. 26. limt2(t2+5t+7) Problem 25E: Evaluating limits Evaluate the following limits. 27. limx15x2+6x+18x4 Problem 26E: Evaluating limits Evaluate the following limits. 28. limt3t2103 Problem 27E: Evaluating limits Find the following limits or state that they do not exist. Assume a, b, c, and k... Problem 28E: Evaluating limits Evaluate the following limits. 30. limx2(x2x)5 Problem 29E: Evaluating limits Evaluate the following limits. 31. limx35x4x3 Problem 30E: Evaluating limits Evaluate the following limits. limh0316+3h+4 Problem 31E: Evaluating limits Evaluate the following limits, where c and k are constants. 61. limx2(5x6)3/2 Problem 32E: Evaluating limits Evaluate the following limits, where c and k are constants. 60. limh0100(10h1)11+2 Problem 33E: Other techniques Evaluate the following limits, where a and b are fixed real numbers. 39. limx1x21x1 Problem 34E: Other techniques Evaluate the following limits, where a and b are fixed real numbers. 40.... Problem 35E: Other techniques Evaluate the following limits, where a and b are fixed real numbers. 41.... Problem 36E: Other techniques Evaluate the following limits, where a and b are fixed real numbers. 42.... Problem 37E: Other techniques Evaluate the following limits, where a and b are fixed real numbers. 43.... Problem 38E: Other techniques Evaluate the following limits, where a and b are fixed real numbers. 44.... Problem 39E: Other techniques Evaluate the following limits, where a and b are fixed real numbers. 45.... Problem 40E: Other techniques Evaluate the following limits, where a and b are fixed real numbers. 46.... Problem 41E: Other techniques Evaluate the following limits, where a and b are fixed real numbers. 47. limx9x3x9 Problem 42E Problem 43E: Evaluating limits Find the following limits or state that they do not exist. Assume a, b, c, and k... Problem 44E: Other techniques Evaluate the following limits, where a and b are fixed real numbers. 48.... Problem 45E: Other techniques Evaluate the following limits, where a and b are fixed real numbers. 49.... Problem 46E: Other techniques Evaluate the following limits, where a and b are fixed real numbers. 50.... Problem 47E: Other techniques Evaluate the following limits, where a and b are fixed real numbers. 51.... Problem 48E: Evaluating limits Evaluate the following limits, where c and k are constants. 66. limxcx22cx+c2xc Problem 49E: Evaluating limits Find the following limits or state that they do not exist. Assume a, b, c, and k... Problem 50E: Evaluating limits Evaluate the following limits, where c and k are constants. 62. limx31x2+2x115x3 Problem 51E: Evaluating limits Evaluate the following limits, where c and k are constants. 63. limx110x91x1 Problem 52E: Evaluating limits Evaluate the following limits, where c and k are constants. 64. limx2(1x22x22x) Problem 53E: Evaluating limits Evaluate the following limits, where c and k are constants. 65. limh0(5+h)225h Problem 54E: Evaluating limits Evaluate the following limits, where c and k are constants. 67.... Problem 55E: Limits involving conjugates Evaluate the following limits. 77. limx1x1x1 Problem 56E: Limits involving conjugates Evaluate the following limits. 78. limx1x14x+53 Problem 57E: Limits involving conjugates Evaluate the following limits. 79. limx43(x4)x+53x+5 Problem 58E Problem 59E: Evaluating limits Find the following limits or state that they do not exist. Assume a, b, c, and k... Problem 60E: Evaluating limits Find the following limits or state that they do not exist. Assume a, b, c, and k... Problem 61E: Evaluating limits Find the following limits or state that they do not exist. Assume a, b, c, and k... Problem 62E: Evaluating limits Find the following limits or state that they do not exist. Assume a, b, c, and k... Problem 63E: Evaluating limits Find the following limits or state that they do not exist. Assume a, b, c, and k... Problem 64E: Evaluating limits Find the following limits or state that they do not exist. Assume a, b, c, and k... Problem 65E Problem 66E: Evaluating limits Find the following limits or state that they do not exist. Assume a, b, c, and k... Problem 67E: Evaluating limits Find the following limits or state that they do not exist. Assume a, b, c, and k... Problem 68E Problem 69E Problem 70E Problem 71E: Explain why or why not Determine whether the following statements are true and give an explanation... Problem 72E Problem 73E: One-sided limits Let f(x)={x2ifx1x+1ifx1. Compute the following limits or state that they do not... Problem 74E: One-sided limits Let f(x)={0ifx525x2if5x53xifx5. Compute the following limits or state that they do... Problem 75E: One-sided limits a. Evaluate limx2+x2. b. Explain why limx2x2 does not exist. Problem 76E: One-sided limits a. Evaluate limx3x32x. b. Explain why limx3+x32x does not exist. Problem 77E: Electric field The magnitude of the electric field at a point x meters from the midpoint of a 0.1 -m... Problem 78E: Torricellis Law A cylindrical tank is filled with water to a depth of 9 meters. At t = 0, a drain in... Problem 79E: Limit of the radius of a cylinder A right circular cylinder with a height of 10 cm and a surface... Problem 80E: A problem from relativity theory Suppose a spaceship of length L0 travels at a high speed v relative... Problem 81E: Applying the Squeeze Theorem a. Show that xxsin1xx, for x 0. b. Illustrate the inequalities in part... Problem 82E: A cosine limit by the Squeeze Theorem It can be shown that 1x22cosx1, for x near 0. a. Illustrate... Problem 83E: A sine limit by the Squeeze Theorem It can be shown that 1x26sinx1, for x near 0. a. Illustrate... Problem 84E: A logarithm limit by the Squeeze Theorem a. Draw a graph to verify that |x| x2 ln x2 |x|, for l x... Problem 85E: Absolute value limit Show that limx0x=0 by first evaluating limx0x and limx0+x. Recall that... Problem 86E: Absolute value limit Show that limxax=a, for any real number. (Hint: Consider the cases a 0 and a ... Problem 87E: Finding a constant Suppose f(x)={x25x+6x3ifx3aifx=3. Determine a value of the constant a for which... Problem 88E: Finding a constant Suppose f(x)={3x+bifx2x2ifx2. Determine a value of the constant b for which... Problem 89E: Finding a constant Suppose g(x)={x25xifx1ax37ifx1. Determine a value of the constant a for which... Problem 90E: Useful factorization formula Calculate the following limits using the factorization formula... Problem 91E: Useful factorization formula Calculate the following limits using the factorization formula... Problem 92E: Useful factorization formula Calculate the following limits using the factorization formula... Problem 93E: Useful factorization formula Calculate the following limits using the factorization formula... Problem 94E: Useful factorization formula Calculate the following limits using the factorization formula... Problem 95E: Slope of a tangent line a. Sketch a graph of y = 2x and carefully draw three secant lines connecting... Problem 96E: Slope of a tangent line a. Sketch a graph of y = 3x and carefully draw four secant lines connecting... Problem 97E: Even function limits Suppose f is an even function where limx1f(x)=5 and limx1+f(x)=6. Find... Problem 98E: Odd function limits Suppose g is an even function where limx1g(x)=5 and limx1+g(x)=6. Find limx1g(x)... Problem 99E: Useful factorization formula Calculate the following limits using the factorization formula... Problem 100E: Evaluate limx16x42x16. Problem 101E: Creating functions satisfying given limit conditions Find functions f and g such that limx1f(x)=0... Problem 102E: Creating functions satisfying given limit conditions Find a function f satisfying limx1(f(x)x1)=2. Problem 103E: Finding constants Find constants b and c in the polynomial p(x) = x2 + bx + c such that... Problem 104E: Limits of composite functions 88. If limx1f(x)=4, find limx1f(x2). Problem 105E: Limits of composite functions 89. Suppose g(x) = f(1 x), for all x, limx1+f(x)=4, and limx1f(x)=6.... Problem 106E: Two trigonometric inequalities Consider the angle in standard position in a unit circle, where 0 ... Problem 107E: Theorem 2.4a Given the polynomial p(x)=bnxn+bn1xn1++b1x+b0 prove that limxap(x)=p(a) for any value... format_list_bulleted