Video Quiz Lecture Notes

Problem 1: Limit Calculation

• Plug in x = 2
• Expression: (2^2 + 7*2 + 6) / (2 + 2)
• Calculation:
  • Numerator: 4 + 14 + 6 = 24
  • Denominator: 2 + 2 = 4
• Limit = 24 / 4 = 6
• Correct Answer: B

Problem 2: Limit with Zero in Denominator

• Direct substitution gives zero in the denominator.
• Factor numerator:
  • Two numbers that multiply to -15 and add to +2 are 5 and -3
  • Factored form: (x + 5)(x - 3)
• Factor denominator:
  • x^2 - 9 = (x - 3)(x + 3)
• Cancel (x - 3):
• Evaluate limit as x approaches 3:
  • Limit = (3 + 5) / (3 + 3) = 8 / 6 = 4 / 3
• Correct Answer: D

Problem 3: Limit of a Complex Fraction

• Multiply top and bottom by common denominator (4x).
• Result:
  • Numerator becomes 4
  • Denominator becomes (x - 4)(4 - x)
• Factor out -1 from (4 - x):
  • Cancel (x - 4):
• Evaluate limit:
  • Limit = -1 / (4*4) = -1 / 16
• Correct Answer: C

Problem 4: Rational Function with Square Root

• Multiply top and bottom by conjugate of numerator.
• Foil the numerator:
  • Result: x + 4√x - 16
• Denominator: (x - 16)(√x + 4)
• Cancel (x - 16):
• Limit as x approaches 16:
  • Limit = 1 / (√16 + 4) = 1 / 8
• Correct Answer: A

Problem 5: Limit with Indeterminate Form

• Direct substitution gives 0 / 0 (indeterminate).
• Check left side limit (as x approaches 7 from left):
  • Using 6.9 gives limit = -1
• Check right side limit (as x approaches 7 from right):
  • Using 7.1 gives limit = 1
• Left and right limits do not match.
• Limit does not exist; Correct Answer: E

Problem 6: Limit of Trigonometric Function

• Replace tangent: tan(3x) = sin(3x) / cos(3x).
• Rewrite:
  • Limit = (3 / 5) * (sin(3x) / 3x)
• Use limit formula:
  • lim (sin y / y) = 1 as y approaches 0.
• Evaluate:
  • Limit = 1 * 1 * (3 / 5) = 3 / 5
• Correct Answer: B

Problem 7: Horizontal Asymptote

• Find limit as x approaches infinity of (5x + 8x^2) / (3x + 2x^2 + 5).
• As x becomes large, focus on leading terms:
  • Limit becomes 8x^2 / 2x^2 = 8 / 2 = 4
• Horizontal asymptote: y = 4; Correct Answer: E

Problem 8: Squeeze Theorem

• Sine function oscillates between -1 and 1.
• As x approaches 0, sin(1/x) oscillates quickly.
• Multiply by x:
  • -x < x*sin(1/x) < x
• Apply squeeze theorem:
  • Limits of -x and x as x approaches 0 are both 0.
• Therefore, limit of x*sin(1/x) = 0.
• Correct Answer: B

Problem 9: Intermediate Value Theorem

• Conditions: f must be continuous on [a, b] and f(a) ≠ f(b).
• Find f(0) and f(2):
  • f(0) = -5; f(2) = 7
• k = 0 is between -5 and 7.
• Set f(c) = 0:
  • Factor: 0 = x^2 + 4x - 5; solutions: x = -5, x = 1
• c = 1 is in [0, 2].
• Correct Answer: D

Problem 10: Continuity at x = 2

• Set functions equal: 7x^2 + cx = 2x^3 + 5c + 3.
• Replace x with 2 and solve for c:
  • 28 = 16 + 3 + 3c
  • 9 = 3c; c = 3
• Correct Answer: C

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These notes summarize the key points and solutions for each limit problem presented in the video quiz.