Problem 1E: How is limxaf(x) calculated if f is a polynomial function? Problem 2E Problem 3E: For what values of a does limxar(x)=r(a) if r is a rational function? Problem 4E Problem 5E Problem 6E Problem 7E: Suppose p and q are polynomials. If limx0p(x)q(x)=10 and q(0) = 2, find p(0). Problem 8E: Suppose limx2f(x)=limx2h(x)=5. Find limx2g(x), where f(x) g(x) h(x), for all x. Problem 9E Problem 10E: Suppose f(x)={4ifx3x+2ifx3. Compute limx3f(x) and limx3+f(x). Problem 11E: Limits of linear functions Evaluate the following limits. 11. limx4(3x7) Problem 12E: Limits of linear functions Evaluate the following limits. 12. limx1(2x+5) Problem 13E: Limits of linear functions Evaluate the following limits. 13. limx95x Problem 14E: Limits of linear functions Evaluate the following limits. 14. limx2(3x) Problem 15E: Limits of linear functions Evaluate the following limits. 15. limx64 Problem 16E: Limits of linear functions Evaluate the following limits. 16. limx5 Problem 17E: Applying limit laws Assume limx1f(x)=8, limx1g(x)=3, and limx1h(x)=2. Compute the following limits... Problem 18E: Applying limit laws Assume limx1f(x)=8, limx1g(x)=3, and limx1h(x)=2. Compute the following limits... Problem 19E: Applying limit laws Assume limx1f(x)=8, limx1g(x)=3, and limx1h(x)=2. Compute the following limits... Problem 20E: Applying limit laws Assume limx1f(x)=8, limx1g(x)=3, and limx1h(x)=2. Compute the following limits... Problem 21E: Applying limit laws Assume limx1f(x)=8, limx1g(x)=3, and limx1h(x)=2. Compute the following limits... Problem 22E: Applying limit laws Assume limx1f(x)=8, limx1g(x)=3, and limx1h(x)=2. Compute the following limits... Problem 23E: Applying limit laws Assume limx1f(x)=8, limx1g(x)=3, and limx1h(x)=2. Compute the following limits... Problem 24E: Applying limit laws Assume limx1f(x)=8, limx1g(x)=3, and limx1h(x)=2. Compute the following limits... Problem 25E: Evaluating limits Evaluate the following limits. 25. limx1(2x33x2+4x+5) Problem 26E: Evaluating limits Evaluate the following limits. 26. limt2(t2+5t+7) Problem 27E: Evaluating limits Evaluate the following limits. 27. limx15x2+6x+18x4 Problem 28E: Evaluating limits Evaluate the following limits. 28. limt3t2103 Problem 29E: Evaluating limits Evaluate the following limits. 29. limb23b4b+11 Problem 30E: Evaluating limits Evaluate the following limits. 30. limx2(x2x)5 Problem 31E: Evaluating limits Evaluate the following limits. 31. limx35x4x3 Problem 32E: Evaluating limits Evaluate the following limits. limh0316+3h+4 Problem 33E: One-sided limits Let f(x)={x2ifx1x+1ifx1. Compute the following limits or state that they do not... Problem 34E: One-sided limits Let f(x)={0ifx525x2if5x53xifx5. Compute the following limits or state that they do... Problem 35E: One-sided limits a. Evaluate limx2+x2. b. Explain why limx2x2 does not exist. Problem 36E: One-sided limits a. Evaluate limx3x32x. b. Explain why limx3+x32x does not exist. Problem 37E: Absolute value limit Show that limx0x=0 by first evaluating limx0x and limx0+x. Recall that... Problem 38E: Absolute value limit Show that limxax=a, for any real number. (Hint: Consider the cases a 0 and a ... Problem 39E: Other techniques Evaluate the following limits, where a and b are fixed real numbers. 39. limx1x21x1 Problem 40E: Other techniques Evaluate the following limits, where a and b are fixed real numbers. 40.... Problem 41E: Other techniques Evaluate the following limits, where a and b are fixed real numbers. 41.... Problem 42E: Other techniques Evaluate the following limits, where a and b are fixed real numbers. 42.... Problem 43E: Other techniques Evaluate the following limits, where a and b are fixed real numbers. 43.... Problem 44E: Other techniques Evaluate the following limits, where a and b are fixed real numbers. 44.... Problem 45E: Other techniques Evaluate the following limits, where a and b are fixed real numbers. 45.... Problem 46E: Other techniques Evaluate the following limits, where a and b are fixed real numbers. 46.... Problem 47E: Other techniques Evaluate the following limits, where a and b are fixed real numbers. 47. limx9x3x9 Problem 48E: Other techniques Evaluate the following limits, where a and b are fixed real numbers. 48.... Problem 49E: Other techniques Evaluate the following limits, where a and b are fixed real numbers. 49.... Problem 50E: Other techniques Evaluate the following limits, where a and b are fixed real numbers. 50.... Problem 51E: Other techniques Evaluate the following limits, where a and b are fixed real numbers. 51.... Problem 52E Problem 53E: Slope of a tangent line a. Sketch a graph of y = 2x and carefully draw three secant lines connecting... Problem 54E Problem 55E: Applying the Squeeze Theorem a. Show that xxsin1xx, for x 0. b. Illustrate the inequalities in part... Problem 56E: A cosine limit by the Squeeze Theorem It can be shown that 1x22cosx1, for x near 0. a. Illustrate... Problem 57E: A sine limit by the Squeeze Theorem It can be shown that 1x26sinx1, for x near 0. a. Illustrate... Problem 58E: A logarithm limit by the Squeeze Theorem a. Draw a graph to verify that |x| x2 ln x2 |x|, for l x... Problem 59E: Explain why or why not Determine whether the following statements are true and give an explanation... Problem 60E: Evaluating limits Evaluate the following limits, where c and k are constants. 60. limh0100(10h1)11+2 Problem 61E: Evaluating limits Evaluate the following limits, where c and k are constants. 61. limx2(5x6)3/2 Problem 62E: Evaluating limits Evaluate the following limits, where c and k are constants. 62. limx31x2+2x115x3 Problem 63E: Evaluating limits Evaluate the following limits, where c and k are constants. 63. limx110x91x1 Problem 64E: Evaluating limits Evaluate the following limits, where c and k are constants. 64. limx2(1x22x22x) Problem 65E: Evaluating limits Evaluate the following limits, where c and k are constants. 65. limh0(5+h)225h Problem 66E: Evaluating limits Evaluate the following limits, where c and k are constants. 66. limxcx22cx+c2xc Problem 67E: Evaluating limits Evaluate the following limits, where c and k are constants. 67.... Problem 68E: Finding a constant Suppose f(x)={3x+bifx2x2ifx2. Determine a value of the constant b for which... Problem 69E: Finding a constant Suppose g(x)={x25xifx1ax37ifx1. Determine a value of the constant a for which... Problem 70E: Useful factorization formula Calculate the following limits using the factorization formula... Problem 71E: Useful factorization formula Calculate the following limits using the factorization formula... Problem 72E: Useful factorization formula Calculate the following limits using the factorization formula... Problem 73E: Useful factorization formula Calculate the following limits using the factorization formula... Problem 74E: Useful factorization formula Calculate the following limits using the factorization formula... Problem 75E: Useful factorization formula Calculate the following limits using the factorization formula... Problem 76E Problem 77E: Limits involving conjugates Evaluate the following limits. 77. limx1x1x1 Problem 78E: Limits involving conjugates Evaluate the following limits. 78. limx1x14x+53 Problem 79E: Limits involving conjugates Evaluate the following limits. 79. limx43(x4)x+53x+5 Problem 80E: Limits involving conjugates Evaluate the following limits. 80. limx0xcx+11, where c is a nonzero... Problem 81E: Creating functions satisfying given limit conditions Find functions f and g such that limx1f(x)=0... Problem 82E: Creating functions satisfying given limit conditions Find a function f satisfying limx1(f(x)x1)=2. Problem 83E: Finding constants Find constants b and c in the polynomial p(x) = x2 + bx + c such that... Problem 84E: A problem from relativity theory Suppose a spaceship of length L0 travels at a high speed v relative... Problem 85E: Limit of the radius of a cylinder A right circular cylinder with a height of 10 cm and a surface... Problem 86E: Torricellis Law A cylindrical tank is filled with water to a depth of 9 meters. At t = 0, a drain in... Problem 87E Problem 88E: Limits of composite functions 88. If limx1f(x)=4, find limx1f(x2). Problem 89E Problem 90E: Two trigonometric inequalities Consider the angle in standard position in a unit circle, where 0 ... Problem 91E format_list_bulleted