Limit Calculation

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Answer Key & Explanations

Limit Calculation · Grade 12 · Worksheet 3

1. A pharmaceutical company is modeling the concentration of a new drug in a patient's bloodstream over time. The concentration function is given by C(t) = (3t^2 - 12)/(t^2 - 4) milligrams per liter, where t is time in hours after administration. As time approaches 2 hours, the function appears to approach an indeterminate form. Determine what concentration the drug approaches as t → 2 hours by simplifying the function algebraically. Answer: 3 Solution: C(t) = (3t^2 - 12) / (t^2 - 4) and we want the limit as t → 2. If we plug in t = 2: Numerator: 3*(2)^2 - 12 = 3*4 - 12 = 12 - 12 = 0 Denominator: (2)^2 - 4 = 4 - 4 = 0 So we get 0/0, which is indeterminate.
   Full step-by-step solution

   Let's go step-by-step. We are given: C(t) = (3t^2 - 12) / (t^2 - 4) and we want the limit as t → 2. --- **Step 1: Direct substitution** If we plug in t = 2: Numerator: 3*(2)^2 - 12 = 3*4 - 12 = 12 - 12 = 0 Denominator: (2)^2 - 4 = 4 - 4 = 0 So we get 0/0, which is indeterminate. This means we need to simplify algebraically. --- **Step 2: Factor numerator and denominator** Numerator: 3t^2 - 12 = 3(t^2 - 4) Denominator: t^2 - 4 = (t - 2)(t + 2) But note: t^2 - 4 appears in numerator and denominator: C(t) = [3(t^2 - 4)] / (t^2 - 4) --- **Step 3: Cancel common factor** For t ≠ 2, t^2 - 4 ≠ 0, so we can cancel: C(t) = 3 --- **Step 4: Interpret the simplification** After canceling, C(t) = 3 for all t except t = 2 or t = -2 (where it is undefined originally). So as t approaches 2, C(t) approaches 3. --- **Final Answer:** 3

2. Hana is an environmental scientist studying the rate of pollutant decay in a lake. The concentration of the pollutant (in parts per million) is modeled by the function C(x) = (4x^2 - 36)/(x - 3) for x ≠ 3, where x represents time in days after a chemical spill. To determine the immediate concentration at the critical time x = 3 days, Hana needs to find what value the concentration approaches as time gets arbitrarily close to 3 days from both sides. What is the limit of C(x) as x approaches 3? Answer: 24 Solution: Write the original function: C(x) = (4x^2 - 36)/(x - 3). Factor the numerator: 4x^2 - 36 = 4(x^2 - 9) = 4(x - 3)(x + 3). Rewrite the function: C(x) = [4(x - 3)(x + 3)]/(x - 3).
   Full step-by-step solution

   Step 1: Write the original function: C(x) = (4x^2 - 36)/(x - 3). Step 2: Factor the numerator: 4x^2 - 36 = 4(x^2 - 9) = 4(x - 3)(x + 3). Step 3: Rewrite the function: C(x) = [4(x - 3)(x + 3)]/(x - 3). Step 4: Cancel the common factor (x - 3) (valid for x ≠ 3): C(x) = 4(x + 3). Step 5: Evaluate the limit as x approaches 3: lim(x→3) 4(x + 3) = 4(3 + 3) = 4(6) = 24. The answer is 24.

3. lim_(x→∞) (5x^3 - 2x^2 + 7)/(3x^3 + 4x - 1) = ? Answer: 5/3 Solution: Identify the highest power of x in both numerator and denominator. Both are degree 3.
   Full step-by-step solution

   Step 1: Identify the highest power of x in both numerator and denominator. Both are degree 3. Step 2: Divide every term by x^3: (5x^3/x^3 - 2x^2/x^3 + 7/x^3)/(3x^3/x^3 + 4x/x^3 - 1/x^3) Step 3: Simplify: (5 - 2/x + 7/x^3)/(3 + 4/x^2 - 1/x^3) Step 4: As x approaches infinity, all terms with x in the denominator approach 0: (5 - 0 + 0)/(3 + 0 - 0) Step 5: The limit equals 5/3 The answer is 5/3.

4. A civil engineer is designing a suspension bridge where the cable's shape is modeled by the function f(x) = (x³ - 8)/(x - 2) for x ≠ 2. To ensure structural integrity at the critical support point x = 2, she needs to determine what height the cable approaches as it gets closer to this point. What is the limit of f(x) as x approaches 2? Answer: 12 Solution: Identify the function: f(x) = (x³ - 8)/(x - 2) Recognize that x³ - 8 is a difference of cubes: x³ - 8 = x³ - 2³ Apply the difference of cubes formula: a³ - b³ = (a - b)(a² + ab + b²) Factor: x³ - 8 = (x - 2)(x² + 2x + 4) Simplify the function: f(x) = [(x - 2)(x² + 2x + 4)]/(x - 2) Cancel the…
   Full step-by-step solution

   Step 1: Identify the function: f(x) = (x³ - 8)/(x - 2) Step 2: Recognize that x³ - 8 is a difference of cubes: x³ - 8 = x³ - 2³ Step 3: Apply the difference of cubes formula: a³ - b³ = (a - b)(a² + ab + b²) Step 4: Factor: x³ - 8 = (x - 2)(x² + 2x + 4) Step 5: Simplify the function: f(x) = [(x - 2)(x² + 2x + 4)]/(x - 2) Step 6: Cancel the common factor (x - 2): f(x) = x² + 2x + 4 for x ≠ 2 Step 7: Evaluate the limit as x approaches 2: lim(x→2) f(x) = 2² + 2(2) + 4 Step 8: Calculate: 4 + 4 + 4 = 12 The answer is 12.

5. A biomedical engineer is modeling the concentration of a new drug in a patient's bloodstream over time. The concentration function is given by C(t) = (t² - 4)/(t - 2) for t ≠ 2, where t is measured in hours. To understand the drug's behavior at the critical time t = 2 hours, the engineer needs to determine the limit of C(t) as t approaches 2. What is this limit value? Answer: 4 Solution: Write down the function. C(t) = (t² - 4)/(t - 2) for t ≠ 2. Direct substitution.
   Full step-by-step solution

   Let's find the limit of C(t) as t approaches 2. Step 1: Write down the function. C(t) = (t² - 4)/(t - 2) for t ≠ 2. Step 2: Direct substitution. If we try to plug in t = 2 directly, we get: Numerator: (2)² - 4 = 4 - 4 = 0 Denominator: 2 - 2 = 0 This gives 0/0, which is an indeterminate form. This means we need to simplify the function. Step 3: Factor the numerator. Notice that t² - 4 is a difference of squares. t² - 4 = (t - 2)(t + 2) Step 4: Rewrite the function. C(t) = (t - 2)(t + 2)/(t - 2) Step 5: Simplify the function. Since t ≠ 2, the term (t - 2) in the numerator and denominator cancels out. C(t) = t + 2, for t ≠ 2. Step 6: Take the limit. The limit as t approaches 2 of C(t) is now the limit as t approaches 2 of (t + 2). lim (t→2) (t + 2) = 2 + 2 = 4. Step 7: Conclusion. Even though the original function is undefined at t = 2, the values of C(t) get arbitrarily close to 4 as t gets close to 2. Therefore, the limit is 4. ANSWER: 4

6. lim_(x→∞) (4x⁴ - 3x³ + 7x)/(2x⁴ + 5x² - 1) = ? Answer: 2 Solution: Identify the highest power of x in both numerator and denominator. Both are degree 4. Divide every term by x⁴, the highest power.
   Full step-by-step solution

   Step 1: Identify the highest power of x in both numerator and denominator. Both are degree 4. Step 2: Divide every term by x⁴, the highest power. Step 3: The expression becomes: (4 - 3/x + 7/x³)/(2 + 5/x² - 1/x⁴) Step 4: As x approaches infinity, all terms with x in the denominator approach 0. Step 5: The limit simplifies to: (4 - 0 + 0)/(2 + 0 - 0) = 4/2 = 2 The answer is 2.

7. lim_(x→∞) (4x³ - 7x² + 2)/(5x³ + 3x - 1) = ? Answer: 4/5 Solution: Identify the highest power of x in both numerator and denominator. Both are degree 3. Divide every term by x³, the highest power.
   Full step-by-step solution

   Step 1: Identify the highest power of x in both numerator and denominator. Both are degree 3. Step 2: Divide every term by x³, the highest power. Step 3: The expression becomes (4 - 7/x + 2/x³)/(5 + 3/x² - 1/x³). Step 4: As x approaches infinity, terms with x in the denominator approach 0. Step 5: The limit simplifies to (4 - 0 + 0)/(5 + 0 - 0) = 4/5. The answer is 4/5.