Limits Concept

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Answer Key & Explanations

Limits Concept · Grade 12 · Worksheet 3

1. From the table: x: 3.9, 3.99, 3.999, 4.001, 4.01, 4.1 f(x): 7.8, 7.98, 7.998, 8.002, 8.02, 8.2 lim(x→4) f(x) = ? Answer: 8 Solution: Examine the f(x) values as x approaches 4 from the left (x = 3.9, 3.99, 3.999).
   Full step-by-step solution

   Step 1: Examine the f(x) values as x approaches 4 from the left (x = 3.9, 3.99, 3.999). - When x = 3.9, f(x) = 7.8 - When x = 3.99, f(x) = 7.98 - When x = 3.999, f(x) = 7.998 As x gets closer to 4 from the left, f(x) gets closer to 8. Step 2: Examine the f(x) values as x approaches 4 from the right (x = 4.001, 4.01, 4.1). - When x = 4.001, f(x) = 8.002 - When x = 4.01, f(x) = 8.02 - When x = 4.1, f(x) = 8.2 As x gets closer to 4 from the right, f(x) also gets closer to 8. Step 3: Since the left-hand limit (as x→4⁻) and the right-hand limit (as x→4⁺) both approach 8, the limit exists and is equal to 8. The answer is 8.

2. A marine biologist is studying the population growth of a coral reef ecosystem. The population P(t) of a particular coral species after t years is modeled by the function P(t) = (2t² - 8)/(t - 2). When analyzing the population behavior as time approaches 2 years, she notices the function appears undefined at exactly t = 2. What value does the coral population approach as time gets closer and closer to 2 years? Answer: 8 Solution: Start with the function P(t) = (2t² - 8)/(t - 2) Factor the numerator: 2t² - 8 = 2(t² - 4) = 2(t - 2)(t + 2) Rewrite the function: P(t) = [2(t - 2)(t + 2)]/(t - 2) Cancel the common factor (t - 2) from numerator and denominator: P(t) = 2(t + 2) Evaluate the simplified function as t approaches 2:…
   Full step-by-step solution

   Step 1: Start with the function P(t) = (2t² - 8)/(t - 2) Step 2: Factor the numerator: 2t² - 8 = 2(t² - 4) = 2(t - 2)(t + 2) Step 3: Rewrite the function: P(t) = [2(t - 2)(t + 2)]/(t - 2) Step 4: Cancel the common factor (t - 2) from numerator and denominator: P(t) = 2(t + 2) Step 5: Evaluate the simplified function as t approaches 2: P(t) approaches 2(2 + 2) = 2(4) = 8 Step 6: Therefore, as time approaches 2 years, the coral population approaches 8 thousand individuals. The answer is 8.

3. A pharmaceutical company is modeling the concentration of a new medication in the bloodstream over time. The concentration function is given by C(t) = (3t^2 - 12) / (t^2 - 4) for t > 2 hours. As time continues indefinitely, what value does the drug concentration approach in the bloodstream? Answer: 3 Solution: C(t) = (3t^2 - 12) / (t^2 - 4) for t > 2. As t becomes very large, both the numerator and denominator become dominated by the highest power of t, which is t^2.
   Full step-by-step solution

   Let's solve this step by step. We are given the concentration function: C(t) = (3t^2 - 12) / (t^2 - 4) for t > 2. --- **Step 1: Identify what happens as t → ∞** As t becomes very large, both the numerator and denominator become dominated by the highest power of t, which is t^2. --- **Step 2: Factor numerator and denominator** Factor numerator: 3t^2 - 12 = 3(t^2 - 4) Factor denominator: t^2 - 4 = (t - 2)(t + 2) but we don't need the full factoring for the limit. Actually, notice: C(t) = (3t^2 - 12) / (t^2 - 4) = 3(t^2 - 4) / (t^2 - 4) --- **Step 3: Simplify the function** For t > 2, t^2 - 4 ≠ 0, so we can cancel: C(t) = 3(t^2 - 4) / (t^2 - 4) = 3 --- **Step 4: Interpret the result** The function simplifies exactly to C(t) = 3 for all t > 2 (except at points where denominator is zero, but t > 2 avoids t = 2). So as t → ∞, C(t) = 3 exactly. --- **Step 5: Conclusion** The concentration approaches 3 mg/L (or whatever units) as time continues indefinitely. --- **Final answer:** 3

4. Mere is a marine biologist studying the oxygen concentration in a lake. She models the oxygen concentration (in mg/L) after t hours of sunlight using the function O(t) = (2t^2 - 6t - 20)/(t - 5). She notices the function is undefined at exactly t = 5 hours, but she needs to know what oxygen concentration the lake approaches as sunlight time gets very close to 5 hours. What value does the oxygen concentration approach? Answer: 14 Solution: Start with the function O(t) = (2t^2 - 6t - 20)/(t - 5) Factor the numerator: 2t^2 - 6t - 20 = 2(t^2 - 3t - 10) Factor the quadratic: t^2 - 3t - 10 = (t - 5)(t + 2) So the numerator is 2(t - 5)(t + 2) Rewrite the function: O(t) = [2(t - 5)(t + 2)]/(t - 5) Cancel the common factor (t - 5) for t ≠…
   Full step-by-step solution

   Step 1: Start with the function O(t) = (2t^2 - 6t - 20)/(t - 5) Step 2: Factor the numerator: 2t^2 - 6t - 20 = 2(t^2 - 3t - 10) Step 3: Factor the quadratic: t^2 - 3t - 10 = (t - 5)(t + 2) Step 4: So the numerator is 2(t - 5)(t + 2) Step 5: Rewrite the function: O(t) = [2(t - 5)(t + 2)]/(t - 5) Step 6: Cancel the common factor (t - 5) for t ≠ 5: O(t) = 2(t + 2) Step 7: As t approaches 5, O(t) approaches 2(5 + 2) Step 8: Calculate: 2(7) = 14 The answer is 14.

5. A particle moves along a straight line with its position given by the function s(t) = (t^3 - 8)/(t - 2) for t ≠ 2. Determine the limit of s(t) as t approaches 2, which represents the instantaneous position the particle would approach at that moment. Answer: 12 Solution: Write down the function. s(t) = (t^3 - 8)/(t - 2) for t ≠ 2. Direct substitution.
   Full step-by-step solution

   Let's find the limit of s(t) as t approaches 2. Step 1: Write down the function. s(t) = (t^3 - 8)/(t - 2) for t ≠ 2. Step 2: Direct substitution. If we try to plug in t = 2 directly: Numerator: 2^3 - 8 = 8 - 8 = 0 Denominator: 2 - 2 = 0 We get 0/0, which is an indeterminate form. This means we need to simplify the function. Step 3: Factor the numerator. We notice t^3 - 8 is a difference of cubes. Recall: a^3 - b^3 = (a - b)(a^2 + ab + b^2) Here, a = t and b = 2. So t^3 - 8 = t^3 - 2^3 = (t - 2)(t^2 + 2t + 4) Step 4: Simplify the function. s(t) = [(t - 2)(t^2 + 2t + 4)]/(t - 2) Since t ≠ 2, we can cancel the (t - 2) factors. This gives: s(t) = t^2 + 2t + 4 for t ≠ 2. Step 5: Take the limit. Now we can find the limit as t approaches 2: lim (t→2) s(t) = lim (t→2) (t^2 + 2t + 4) Step 6: Evaluate the simplified expression. Plug in t = 2: 2^2 + 2×2 + 4 = 4 + 4 + 4 = 12 Therefore, the limit of s(t) as t approaches 2 is 12. Explanation: Even though s(t) is undefined exactly at t = 2, as t gets very close to 2, the position s(t) gets arbitrarily close to 12. This represents the instantaneous position the particle would approach at that moment.

6. Liam is designing a water tank with a capacity of 1000 liters. The tank's height decreases over time due to sediment accumulation at a rate modeled by h(t) = 2 - 0.1e^(-0.05t) meters, where t is time in years. As time approaches infinity, what height does the water level approach in the tank? Answer: 2 meters Solution: h(t) = 2 - 0.1 * e^(-0.05t) meters where t is in years.
   Full step-by-step solution

   Let's go step-by-step. We are given the height function: h(t) = 2 - 0.1 * e^(-0.05t) meters where t is in years. We want the height as time approaches infinity, so we take the limit: limit as t → ∞ of h(t) = limit as t → ∞ of [2 - 0.1 * e^(-0.05t)] Step 1: Understand the exponential term. e^(-0.05t) can be written as 1 / e^(0.05t). As t → ∞, e^(0.05t) → ∞, so 1 / e^(0.05t) → 0. Step 2: Substitute the limiting behavior into h(t). h(t) = 2 - 0.1 * [something approaching 0] So h(t) → 2 - 0.1 * 0 Step 3: Simplify. 2 - 0 = 2 Step 4: Conclusion. As time approaches infinity, the height approaches 2 meters. This makes sense because the sediment accumulation reduces the height by an amount 0.1 * e^(-0.05t), which becomes negligible for large t, leaving only the 2 meters. ANSWER: 2 meters

7. A pharmaceutical company is modeling the concentration of a new drug in a patient's bloodstream using the function C(t) = (3t² - 12)/(t - 2), where t is time in hours after administration. The researchers need to determine what concentration the drug approaches as time gets very close to 2 hours, when the function appears undefined. What is this limiting concentration value? Answer: 12 Solution: C(t) = (3t² - 12)/(t - 2) At t = 2, the denominator becomes 2 - 2 = 0, so the function is undefined. We need to find the limit as t approaches 2.
   Full step-by-step solution

   Let's find the limiting concentration as t approaches 2 hours. We have: C(t) = (3t² - 12)/(t - 2) Step 1: Identify the problem At t = 2, the denominator becomes 2 - 2 = 0, so the function is undefined. We need to find the limit as t approaches 2. Step 2: Factor the numerator Notice that 3t² - 12 = 3(t² - 4) And t² - 4 = (t - 2)(t + 2) by the difference of squares formula. So the numerator becomes: 3t² - 12 = 3(t - 2)(t + 2) Step 3: Simplify the function C(t) = [3(t - 2)(t + 2)]/(t - 2) Since t is approaching 2 but not equal to 2, we can cancel the (t - 2) factor from numerator and denominator. This gives: C(t) = 3(t + 2) Step 4: Evaluate the limit As t approaches 2, C(t) approaches 3(2 + 2) = 3 × 4 = 12 Therefore, the drug concentration approaches 12 as time gets very close to 2 hours. ANSWER: 12