Limit Calculation

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Answer Key & Explanations

Limit Calculation · Grade 12 · Worksheet 1

1. lim_{x→0} (sin(3x))/(2x) = ? Answer: 1.5 Solution: The problem is lim_{x→0} (sin(3x))/(2x). We know the standard limit: lim_{θ→0} (sin(θ))/(θ) = 1. To use this, we need the denominator to match the argument of sine.
   Full step-by-step solution

   Step 1: The problem is lim_{x→0} (sin(3x))/(2x). Step 2: We know the standard limit: lim_{θ→0} (sin(θ))/(θ) = 1. Step 3: To use this, we need the denominator to match the argument of sine. Multiply numerator and denominator by 3/3: (sin(3x))/(2x) = (3/2) × (sin(3x))/(3x). Step 4: Now we have lim_{x→0} (3/2) × (sin(3x))/(3x). Step 5: As x→0, 3x→0, so (sin(3x))/(3x) → 1. Step 6: Therefore, the limit equals (3/2) × 1 = 3/2. Step 7: 3/2 = 1.5 The answer is 1.5.

2. Tane is a marine biologist studying the population growth of a rare species of fish in a protected lagoon. The population size, in hundreds, is modeled by the function P(t) = (t^3 - 27)/(t - 3) for t ≠ 3, where t represents time in years since the study began. Due to a data collection gap, the population at exactly t = 3 years was not recorded. Tane needs to determine what population value the function approaches as time gets arbitrarily close to 3 years to complete his report. What is the limit of P(t) as t approaches 3? Answer: 27 Solution: Write the limit: lim(t→3) (t^3 - 27)/(t - 3) Factor the numerator using difference of cubes: t^3 - 27 = (t - 3)(t^2 + 3t + 9) Rewrite the function: P(t) = [(t - 3)(t^2 + 3t + 9)]/(t - 3) Cancel the common factor (t - 3): P(t) = t^2 + 3t + 9 for t ≠ 3 Evaluate the simplified function at t = 3:…
   Full step-by-step solution

   Step 1: Write the limit: lim(t→3) (t^3 - 27)/(t - 3) Step 2: Factor the numerator using difference of cubes: t^3 - 27 = (t - 3)(t^2 + 3t + 9) Step 3: Rewrite the function: P(t) = [(t - 3)(t^2 + 3t + 9)]/(t - 3) Step 4: Cancel the common factor (t - 3): P(t) = t^2 + 3t + 9 for t ≠ 3 Step 5: Evaluate the simplified function at t = 3: 3^2 + 3(3) + 9 = 9 + 9 + 9 = 27 The answer is 27.

3. A civil engineer is designing a suspension bridge where the cable's shape follows the function f(x) = (x³ - 8)/(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 = (x - 2)(x² + 2x + 4) Rewrite the function: f(x) = [(x - 2)(x² + 2x + 4)]/(x - 2) Cancel the common factor (x - 2) for x ≠ 2: f(x) = x² + 2x + 4 Evaluate the limit as x approaches 2: lim(x→2)…
   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 = (x - 2)(x² + 2x + 4) Step 3: Rewrite the function: f(x) = [(x - 2)(x² + 2x + 4)]/(x - 2) Step 4: Cancel the common factor (x - 2) for x ≠ 2: f(x) = x² + 2x + 4 Step 5: Evaluate the limit as x approaches 2: lim(x→2) f(x) = (2)² + 2(2) + 4 Step 6: Calculate: 4 + 4 + 4 = 12 The answer is 12.

4. A biomedical engineer is modeling the concentration of a new medication in a patient's bloodstream over time. The concentration function is C(t) = (t² - 4)/(t - 2) for t ≠ 2, where t is measured in hours. The engineer needs to determine what concentration the medication approaches as time approaches 2 hours to ensure safe dosage levels. What is the limiting concentration as t approaches 2 hours? Answer: 4 Solution: C(t) = (t² - 4)/(t - 2) for t ≠ 2. We want the limit as t approaches 2.
   Full step-by-step solution

   Let's solve step by step. We are given: C(t) = (t² - 4)/(t - 2) for t ≠ 2. We want the limit as t approaches 2. --- **Step 1: Identify the problem at t = 2** If we plug t = 2 directly into C(t), we get: Numerator: (2² - 4) = (4 - 4) = 0 Denominator: (2 - 2) = 0 So we have 0/0, which is indeterminate. That means we need to simplify the function. --- **Step 2: Factor the numerator** t² - 4 is a difference of squares: t² - 4 = (t - 2)(t + 2) So C(t) = (t - 2)(t + 2) / (t - 2) --- **Step 3: Cancel the common factor** For t ≠ 2, (t - 2)/(t - 2) = 1. So C(t) = t + 2, for t ≠ 2. --- **Step 4: Take the limit** As t approaches 2, C(t) approaches 2 + 2 = 4. --- **Step 5: Conclusion** The limiting concentration as t approaches 2 hours is 4.

5. A civil engineer is designing a suspension bridge and needs to analyze the behavior of the cable tension near a critical support point. The tension function is given by T(x) = (2x² - 8)/(x - 2) kilonewtons, where x represents the distance in meters from the support. As the cable approaches the support at x = 2 meters, the function appears undefined. What tension value does the cable approach as x gets arbitrarily close to 2 meters from both sides? Answer: 8 Solution: Identify the limit we need to find: lim(x→2) (2x² - 8)/(x - 2) Factor the numerator: 2x² - 8 = 2(x² - 4) = 2(x - 2)(x + 2) Rewrite the function: T(x) = [2(x - 2)(x + 2)]/(x - 2) Cancel the common factor (x - 2) from numerator and denominator: T(x) = 2(x + 2) for x ≠ 2 Evaluate the simplified…
   Full step-by-step solution

   Step 1: Identify the limit we need to find: lim(x→2) (2x² - 8)/(x - 2) Step 2: Factor the numerator: 2x² - 8 = 2(x² - 4) = 2(x - 2)(x + 2) Step 3: Rewrite the function: T(x) = [2(x - 2)(x + 2)]/(x - 2) Step 4: Cancel the common factor (x - 2) from numerator and denominator: T(x) = 2(x + 2) for x ≠ 2 Step 5: Evaluate the simplified function at x = 2: T(x) = 2(2 + 2) = 2(4) = 8 Step 6: Therefore, as x approaches 2, the tension approaches 8 kilonewtons. The answer is 8.

6. lim_{x→3} (x² - 9)/(x - 3) = ? Answer: 6 Solution: lim_{x→3} (x² - 9)/(x - 3) If we substitute x = 3 into the expression, we get: (3² - 9)/(3 - 3) = (9 - 9)/(0) = 0/0 This is an indeterminate form, so we need to simplify the expression.
   Full step-by-step solution

   Let's solve the limit step by step. We are given: lim_{x→3} (x² - 9)/(x - 3) --- **Step 1: Direct substitution** If we substitute x = 3 into the expression, we get: (3² - 9)/(3 - 3) = (9 - 9)/(0) = 0/0 This is an indeterminate form, so we need to simplify the expression. --- **Step 2: Factor the numerator** Notice that x² - 9 is a difference of squares: x² - 9 = (x - 3)(x + 3) So the expression becomes: (x² - 9)/(x - 3) = (x - 3)(x + 3)/(x - 3) --- **Step 3: Cancel the common factor** For x ≠ 3, we can cancel (x - 3) from numerator and denominator: (x - 3)(x + 3)/(x - 3) = x + 3 --- **Step 4: Take the limit** Now we have: lim_{x→3} (x + 3) This is a simple polynomial, so we substitute x = 3: 3 + 3 = 6 --- **Final Answer:** 6

7. lim_(x→2) (x² - 4)/(x - 2) = ? Answer: 4 Solution: lim_(x→2) (x² - 4)/(x - 2) If we substitute x = 2 into the expression, we get: (2² - 4)/(2 - 2) = (4 - 4)/(0) = 0/0 This is an indeterminate form, so we need to simplify the expression.
   Full step-by-step solution

   Let's solve the limit step-by-step. We are given: lim_(x→2) (x² - 4)/(x - 2) Step 1: Direct substitution If we substitute x = 2 into the expression, we get: (2² - 4)/(2 - 2) = (4 - 4)/(0) = 0/0 This is an indeterminate form, so we need to simplify the expression. Step 2: Factor the numerator Notice that x² - 4 is a difference of squares: x² - 4 = (x - 2)(x + 2) Step 3: Rewrite the expression So (x² - 4)/(x - 2) = [(x - 2)(x + 2)]/(x - 2) For x ≠ 2, we can cancel (x - 2) from numerator and denominator. Step 4: Simplify After canceling, we get: x + 2 Step 5: Take the limit Now we take the limit as x approaches 2: lim_(x→2) (x + 2) = 2 + 2 = 4 Therefore, the limit is 4.