Function End Behavior

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Answer Key & Explanations

Function End Behavior · Grade 12 · Worksheet 2

1. lim(x→-∞) (8x⁶ - 6x⁴ + 2)/(4x⁶ + 10x³ - 2) = ? Answer: 2 Solution: Identify the highest degree terms. Numerator: 8x⁶ (degree 6). Denominator: 4x⁶ (degree 6).
   Full step-by-step solution

   Step 1: Identify the highest degree terms. Numerator: 8x⁶ (degree 6). Denominator: 4x⁶ (degree 6). Step 2: Since degrees are equal, divide both numerator and denominator by x⁶: (8x⁶/x⁶ - 6x⁴/x⁶ + 2/x⁶) / (4x⁶/x⁶ + 10x³/x⁶ - 2/x⁶) = (8 - 6/x² + 2/x⁶) / (4 + 10/x³ - 2/x⁶) Step 3: As x→-∞, terms with x in the denominator approach 0: 6/x² → 0, 2/x⁶ → 0, 10/x³ → 0, -2/x⁶ → 0. Step 4: The limit simplifies to (8 - 0 + 0) / (4 + 0 - 0) = 8/4 = 2. The answer is 2.

2. Aroha, an environmental physicist, is modeling the rate of radioactive decay in a contaminated soil sample. The decay rate R(t) (in becquerels per kilogram) as a function of time t (in years) is given by R(t) = (9t^3 - 12t^2 + 15) / (3t^3 + 6t - 8). As time approaches infinity, what constant decay rate does the model approach? Answer: 3 Solution: Identify the highest power terms in numerator and denominator. Numerator: 9t^3 - 12t^2 + 15 → highest power is t^3 with coefficient 9. Denominator: 3t^3 + 6t - 8 → highest power is t^3 with coefficient 3.
   Full step-by-step solution

   Step 1: Identify the highest power terms in numerator and denominator. Numerator: 9t^3 - 12t^2 + 15 → highest power is t^3 with coefficient 9. Denominator: 3t^3 + 6t - 8 → highest power is t^3 with coefficient 3. Step 2: For rational functions where the degrees of numerator and denominator are equal, the horizontal asymptote (end behavior) is the ratio of the leading coefficients. Step 3: Calculate the ratio: 9 / 3 = 3. Step 4: As t approaches infinity, the lower-degree terms become negligible compared to the t^3 terms, so the function approaches 3. Therefore, as t → ∞, R(t) → 3 becquerels per kilogram.

3. Noah is an environmental scientist modeling the concentration of a pollutant in a river over time after a cleanup effort begins. The concentration C(t) in parts per million (ppm) is given by the function C(t) = (6t^2 - 11t + 1) / (t^2 + 6t - 1), where t represents time in months since the cleanup started. As t approaches infinity, what concentration level does the pollutant approach according to this model? Answer: 6 Solution: Identify the degrees of the numerator and denominator. The numerator is 6t^2 - 11t + 1 (degree 2) and the denominator is t^2 + 6t - 1 (degree 2). They are equal.
   Full step-by-step solution

   Step 1: Identify the degrees of the numerator and denominator. The numerator is 6t^2 - 11t + 1 (degree 2) and the denominator is t^2 + 6t - 1 (degree 2). They are equal. Step 2: For rational functions with equal degrees, the horizontal asymptote is found by taking the ratio of the leading coefficients. Step 3: The leading coefficient of the numerator is 6, and the leading coefficient of the denominator is 1. Step 4: The ratio is 6/1 = 6. Step 5: Therefore, as t approaches infinity, the concentration C(t) approaches 6 ppm. The answer is 6.

4. A biomedical engineer is modeling the concentration of a new drug in a patient's bloodstream over time using the function C(t) = (3t³ - 2t² + 5)/(t³ + 1), where C(t) represents the drug concentration in milligrams per liter and t represents time in hours. As time approaches infinity, what value does the drug concentration approach? Answer: 3 Solution: C(t) = (3t³ - 2t² + 5)/(t³ + 1) Identify the highest power of t in the denominator. The denominator is t³ + 1, so the highest power is t³. Divide both the numerator and denominator by this highest power (t³).
   Full step-by-step solution

   Let's find the limit of C(t) as t approaches infinity. We have: C(t) = (3t³ - 2t² + 5)/(t³ + 1) Step 1: Identify the highest power of t in the denominator. The denominator is t³ + 1, so the highest power is t³. Step 2: Divide both the numerator and denominator by this highest power (t³). Numerator: (3t³ - 2t² + 5)/t³ = 3t³/t³ - 2t²/t³ + 5/t³ = 3 - 2/t + 5/t³ Denominator: (t³ + 1)/t³ = t³/t³ + 1/t³ = 1 + 1/t³ So C(t) becomes: C(t) = (3 - 2/t + 5/t³)/(1 + 1/t³) Step 3: Take the limit as t approaches infinity. As t → ∞: - 2/t → 0 - 5/t³ → 0 - 1/t³ → 0 So we get: lim(t→∞) C(t) = (3 - 0 + 0)/(1 + 0) = 3/1 = 3 Step 4: Conclusion. As time approaches infinity, the drug concentration approaches 3 milligrams per liter. This makes sense because for large values of t, the highest degree terms (3t³ in numerator and t³ in denominator) dominate the behavior of the function, and their ratio is 3.

5. lim_{x→∞} (2x⁴ - 5x³ + 3)/(x⁴ + 7x² - 2) = ? Answer: 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 2, leading coefficient of denominator is 1. Step 4: Therefore, the limit is 2/1 = 2. The answer is 2.

6. A biomedical engineer is modeling the concentration of a new medication in a patient's bloodstream over time using the function C(t) = (3t³ - 2t + 1)/(t³ + 5), where C is concentration in mg/L and t is time in hours. As time approaches infinity, what value does the drug concentration approach, and what does this tell the engineer about the long-term behavior of the medication in the body? Answer: 3 mg/L Solution: To find the limiting value of the concentration function C(t) = (3t³ - 2t + 1)/(t³ + 5) as time approaches infinity, we analyze the behavior of the numerator and denominator for very large values of t.
   Full step-by-step solution

   To find the limiting value of the concentration function C(t) = (3t³ - 2t + 1)/(t³ + 5) as time approaches infinity, we analyze the behavior of the numerator and denominator for very large values of t. Step 1: Identify the highest power of t in the denominator. The denominator is t³ + 5. The highest power of t here is t³. Step 2: Divide every term in both the numerator and the denominator by this highest power. We will divide every term by t³. The numerator is 3t³ - 2t + 1. Dividing each term by t³ gives: 3t³/t³ - (2t)/t³ + 1/t³ = 3 - 2/t² + 1/t³ The denominator is t³ + 5. Dividing each term by t³ gives: t³/t³ + 5/t³ = 1 + 5/t³ So, the function becomes: C(t) = [3 - 2/t² + 1/t³] / [1 + 5/t³] Step 3: Determine what happens to each term as t approaches infinity. As t becomes very large (approaches infinity), any term with t in the denominator will approach zero. - The term 2/t² approaches 0. - The term 1/t³ approaches 0. - The term 5/t³ approaches 0. Step 4: Take the limit by substituting these values. Substitute 0 for all the terms that are divided by a power of t: C(t) = [3 - 0 + 0] / [1 + 0] = 3 / 1 = 3 Conclusion: As time approaches infinity, the drug concentration C(t) approaches 3 mg/L. Interpretation for the Engineer: This result tells the engineer that in the long term, the concentration of the medication in the bloodstream will stabilize at 3 mg/L. It will not keep increasing indefinitely nor drop to zero. This stable concentration is known as the steady-state concentration and is a crucial parameter for determining the correct and safe dosage of the medication over time.

7. Consider the function f(x) = (3x^4 - 2x^3 + 5x - 1)/(2x^4 - x^2 + 7). As x approaches positive infinity, what value does f(x) approach? Answer: 1.5 Solution: To determine the value that f(x) approaches as x approaches positive infinity, we analyze the function: f(x) = (3x^4 - 2x^3 + 5x - 1)/(2x^4 - x^2 + 7) Identify the highest power of x in the denominator.
   Full step-by-step solution

   To determine the value that f(x) approaches as x approaches positive infinity, we analyze the function: f(x) = (3x^4 - 2x^3 + 5x - 1)/(2x^4 - x^2 + 7) Step 1: Identify the highest power of x in the denominator. The denominator is 2x^4 - x^2 + 7. The highest power of x here is x^4. Step 2: Divide every term in the numerator and denominator by x^4. This is the standard approach for limits at infinity of rational functions. We divide each term by x^4: Numerator: (3x^4)/x^4 - (2x^3)/x^4 + (5x)/x^4 - 1/x^4 = 3 - 2/x + 5/x^3 - 1/x^4 Denominator: (2x^4)/x^4 - (x^2)/x^4 + 7/x^4 = 2 - 1/x^2 + 7/x^4 So the function becomes: f(x) = (3 - 2/x + 5/x^3 - 1/x^4) / (2 - 1/x^2 + 7/x^4) Step 3: Evaluate the limit as x approaches positive infinity. As x becomes very large (approaches infinity), terms with x in the denominator approach 0: - 2/x → 0 - 5/x^3 → 0 - 1/x^4 → 0 - 1/x^2 → 0 - 7/x^4 → 0 Step 4: Substitute these limiting values into the expression. After these terms go to 0, we're left with: f(x) → (3 - 0 + 0 - 0) / (2 - 0 + 0) = 3/2 Step 5: Convert to decimal form. 3/2 = 1.5 Therefore, as x approaches positive infinity, f(x) approaches 1.5. This makes sense because for large values of x, the highest degree terms (3x^4 in numerator and 2x^4 in denominator) dominate the behavior of the function, and their ratio is 3/2 = 1.5.