Function End Behavior

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Answer Key & Explanations

Function End Behavior · Grade 12 · Worksheet 3

1. lim(x→∞) (5x⁴ - 3x² + 7)/(2x⁴ + x³ - 8) = ? Answer: 5/2 Solution: Identify the degrees of numerator and denominator. Both are degree 4. For rational functions where degrees are equal, the limit equals the ratio of leading coefficients.
   Full step-by-step solution

   Step 1: Identify the degrees of numerator and denominator. Both are degree 4. Step 2: For rational functions where degrees are equal, the limit equals the ratio of leading coefficients. Step 3: Leading coefficient of numerator is 5, leading coefficient of denominator is 2. Step 4: The limit is 5/2. The answer is 5/2.

2. lim(x→∞) (4x⁵ - 2x³ + 8)/(3x⁵ + 5x² - 1) = ? Answer: 4/3 Solution: Identify the highest degree terms in numerator and denominator Numerator: 4x⁵ Denominator: 3x⁵ For large x values, the lower degree terms become insignificant lim(x→∞) (4x⁵ - 2x³ + 8)/(3x⁵ + 5x² - 1) ≈ lim(x→∞) (4x⁵)/(3x⁵) (4x⁵)/(3x⁵) = 4/3 The limit equals the ratio of the leading coefficients…
   Full step-by-step solution

   Step 1: Identify the highest degree terms in numerator and denominator Numerator: 4x⁵ Denominator: 3x⁵ Step 2: For large x values, the lower degree terms become insignificant lim(x→∞) (4x⁵ - 2x³ + 8)/(3x⁵ + 5x² - 1) ≈ lim(x→∞) (4x⁵)/(3x⁵) Step 3: Simplify the ratio of the leading terms (4x⁵)/(3x⁵) = 4/3 Step 4: The limit equals the ratio of the leading coefficients lim(x→∞) (4x⁵ - 2x³ + 8)/(3x⁵ + 5x² - 1) = 4/3 The answer is 4/3.

3. Charlotte is a pharmaceutical researcher modeling the long-term concentration of a new antibiotic in a patient's bloodstream. The concentration C(t), measured in micrograms per milliliter (mcg/mL), is given by the function C(t) = (9t^3 - 14t^2 + 8) / (3t^3 + 7t - 12), where t represents time in hours after the initial dose. As time increases indefinitely (t → ∞), what concentration level does the antibiotic approach in the patient's bloodstream? Answer: 3 Solution: Identify the degrees of the numerator and denominator. The numerator is 9t^3 - 14t^2 + 8 (degree 3) and the denominator is 3t^3 + 7t - 12 (degree 3). Both are degree 3.
   Full step-by-step solution

   Step 1: Identify the degrees of the numerator and denominator. The numerator is 9t^3 - 14t^2 + 8 (degree 3) and the denominator is 3t^3 + 7t - 12 (degree 3). Both are degree 3. Step 2: Since the degrees are equal, the horizontal asymptote (the value as t → ∞) is the ratio of the leading coefficients. Step 3: The leading coefficient of the numerator is 9. The leading coefficient of the denominator is 3. Step 4: Compute the ratio: 9 / 3 = 3. Step 5: Therefore, as t → ∞, C(t) approaches 3 mcg/mL. The answer is 3.

4. An environmental scientist is modeling the population growth of an endangered species using the function P(t) = (4t^4 - 3t^3 + 2t - 7)/(2t^4 + 5t^2 - 1), where P represents the population in thousands and t represents time in years. As time extends indefinitely into the future, what population level will the species approach according to this model? Answer: 2 Solution: Identify the highest degree terms in numerator and denominator Numerator: 4t^4 Denominator: 2t^4 As t approaches infinity, the lower degree terms (-3t^3, +2t, -7 in numerator and +5t^2, -1 in denominator) become negligible compared to the highest degree terms The function approaches the ratio of…
   Full step-by-step solution

   Step 1: Identify the highest degree terms in numerator and denominator Numerator: 4t^4 Denominator: 2t^4 Step 2: As t approaches infinity, the lower degree terms (-3t^3, +2t, -7 in numerator and +5t^2, -1 in denominator) become negligible compared to the highest degree terms Step 3: The function approaches the ratio of the coefficients of the highest degree terms P(t) ≈ (4t^4)/(2t^4) = 4/2 = 2 Step 4: Therefore, as t approaches infinity, P(t) approaches 2 The answer is 2.

5. lim(x→∞) (3x³ - 4x² + 2)/(2x³ + 5x - 1) = ? Answer: 3/2 Solution: Identify the highest degree terms in numerator and denominator Numerator: 3x³ Denominator: 2x³ For rational functions where degrees are equal, the limit equals the ratio of leading coefficients Ratio = 3/2 Numerator degree: 3 Denominator degree: 3 Since degrees are equal, limit = 3/2 The answer…
   Full step-by-step solution

   Step 1: Identify the highest degree terms in numerator and denominator Numerator: 3x³ Denominator: 2x³ Step 2: For rational functions where degrees are equal, the limit equals the ratio of leading coefficients Ratio = 3/2 Step 3: Verify degrees are equal Numerator degree: 3 Denominator degree: 3 Step 4: Since degrees are equal, limit = 3/2 The answer is 3/2.

6. lim(x→∞) (2x⁴ - 3x² + 7)/(5x⁴ - x³ + 2) = ? Answer: 2/5 Solution: Identify the degrees of numerator and denominator. Both are degree 4. Since degrees are equal, the limit equals the ratio of leading coefficients.
   Full step-by-step solution

   Step 1: Identify the degrees of numerator and denominator. Both are degree 4. Step 2: Since degrees are equal, the limit equals the ratio of leading coefficients. Step 3: Leading coefficient of numerator is 2. Step 4: Leading coefficient of denominator is 5. Step 5: The limit is 2/5. The answer is 2/5.

7. A biomedical engineer is modeling the concentration of a new drug in a patient's bloodstream using the function C(t) = (3t^3 - 2t^2 + 5)/(t^3 - 4), where t represents hours after administration. As time approaches infinity, what value does the drug concentration approach? Answer: 3 Solution: To find the value that C(t) approaches as time approaches infinity, we examine the function: C(t) = (3t^3 - 2t^2 + 5) / (t^3 - 4) Identify the highest power of t in the denominator. The denominator is t^3 - 4.
   Full step-by-step solution

   To find the value that C(t) approaches as time approaches infinity, we examine the function: C(t) = (3t^3 - 2t^2 + 5) / (t^3 - 4) Step 1: Identify the highest power of t in the denominator. The denominator is t^3 - 4. The highest power of t here is t^3. Step 2: Divide every term in the numerator and denominator by this highest power (t^3). This technique helps us see what happens when t becomes very large, because terms like 1/t, 1/t^2, etc., will approach zero. Numerator: 3t^3/t^3 - 2t^2/t^3 + 5/t^3 = 3 - 2/t + 5/t^3 Denominator: t^3/t^3 - 4/t^3 = 1 - 4/t^3 So the function becomes: C(t) = (3 - 2/t + 5/t^3) / (1 - 4/t^3) Step 3: Analyze the limit as t approaches infinity. As t becomes infinitely large (t -> infinity): - The term 2/t approaches 0. - The term 5/t^3 approaches 0. - The term 4/t^3 approaches 0. Substituting these limiting values into our expression: C(t) approaches (3 - 0 + 0) / (1 - 0) = 3/1 = 3 Step 4: Conclusion. Therefore, as time approaches infinity, the drug concentration C(t) approaches the value 3. ANSWER: 3

8. lim(x→-∞) (6x⁶ - 11x⁴ + 1)/(3x⁶ + 7x² - 6) = ? Answer: 2 Solution: Identify the highest degree terms in numerator and denominator. Numerator: 6x⁶ (degree 6). Denominator: 3x⁶ (degree 6).
   Full step-by-step solution

   Step 1: Identify the highest degree terms in numerator and denominator. Numerator: 6x⁶ (degree 6). Denominator: 3x⁶ (degree 6). Step 2: Since the degrees are equal, divide both numerator and denominator by x⁶: (6x⁶/x⁶ - 11x⁴/x⁶ + 1/x⁶) / (3x⁶/x⁶ + 7x²/x⁶ - 6/x⁶) = (6 - 11/x² + 1/x⁶) / (3 + 7/x⁴ - 6/x⁶) Step 3: Evaluate the limit as x→-∞. As x→-∞, 11/x² → 0, 1/x⁶ → 0, 7/x⁴ → 0, 6/x⁶ → 0. Step 4: The limit simplifies to (6 - 0 + 0) / (3 + 0 - 0) = 6/3 = 2. The answer is 2.