Limits Concept

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Answer Key & Explanations

Limits Concept · Grade 12 · Worksheet 2

1. A solid is formed by rotating the region bounded by the curve y = sqrt(x), the x-axis, and the vertical line x = 4 about the x-axis. Using the disk method, describe the volume of this solid as a definite integral. What is the integral expression for the volume? Answer: ∫ from 0 to 4 of π(x) dx Solution: The region is bounded by y = sqrt(x), y = 0 (x-axis), and x = 4. When rotated about the x-axis, this creates a solid where each cross-section perpendicular to the x-axis is a disk.
   Full step-by-step solution

   Step 1: The region is bounded by y = sqrt(x), y = 0 (x-axis), and x = 4. Step 2: When rotated about the x-axis, this creates a solid where each cross-section perpendicular to the x-axis is a disk. Step 3: For the disk method, the volume is ∫ from a to b of π[radius]² dx. Step 4: The radius at any x-value is given by y = sqrt(x). Step 5: So the radius squared is (sqrt(x))² = x. Step 6: The x-values range from 0 to 4. Step 7: Therefore, the volume integral is ∫ from 0 to 4 of π(x) dx. The integral expression for the volume is ∫ from 0 to 4 of π(x) dx.

2. Emma is analyzing the rate at which a chemical reaction produces product. The amount of product P(t) in grams after t minutes is modeled by the function P(t) = (t^2 - 16)/(t - 4). She needs to determine what amount of product the reaction approaches as time gets very close to 4 minutes, even though the function is undefined at exactly t = 4 minutes. Answer: 8 Solution: Identify the function: P(t) = (t^2 - 16)/(t - 4) Factor the numerator: t^2 - 16 = (t - 4)(t + 4) Rewrite the function: P(t) = [(t - 4)(t + 4)]/(t - 4) Cancel the common factor (t - 4) for t ≠ 4: P(t) = t + 4 Find the limit as t approaches 4: lim(t→4) (t + 4) = 4 + 4 = 8 The product amount…
   Full step-by-step solution

   Step 1: Identify the function: P(t) = (t^2 - 16)/(t - 4) Step 2: Factor the numerator: t^2 - 16 = (t - 4)(t + 4) Step 3: Rewrite the function: P(t) = [(t - 4)(t + 4)]/(t - 4) Step 4: Cancel the common factor (t - 4) for t ≠ 4: P(t) = t + 4 Step 5: Find the limit as t approaches 4: lim(t→4) (t + 4) = 4 + 4 = 8 Step 6: The product amount approaches 8 grams as time gets closer to 4 minutes.

3. 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

4. Ava is studying the velocity of a drone as it approaches a landing pad. The drone's velocity in meters per second, as a function of time t in seconds, is given by v(t) = (t^2 - 49)/(t - 7). Ava notices that the function is undefined at exactly t = 7 seconds, but she wants to know what velocity the drone approaches as time gets closer and closer to 7 seconds from either direction. What is the limiting velocity of the drone as t approaches 7 seconds? Answer: 14 Solution: Write the velocity function: v(t) = (t^2 - 49)/(t - 7). Factor the numerator using difference of squares: t^2 - 49 = (t - 7)(t + 7). Rewrite the function: v(t) = [(t - 7)(t + 7)]/(t - 7).
   Full step-by-step solution

   Step 1: Write the velocity function: v(t) = (t^2 - 49)/(t - 7). Step 2: Factor the numerator using difference of squares: t^2 - 49 = (t - 7)(t + 7). Step 3: Rewrite the function: v(t) = [(t - 7)(t + 7)]/(t - 7). Step 4: Cancel the common factor (t - 7) for all t not equal to 7: v(t) = t + 7. Step 5: Find the limit as t approaches 7: lim(t→7) v(t) = lim(t→7) (t + 7) = 7 + 7. Step 6: Calculate the sum: 7 + 7 = 14. The drone's velocity approaches 14 meters per second as t approaches 7 seconds.

5. lim(x→2) (x² - 4)/(x - 2) = ? Answer: 4 Solution: The original limit is lim(x→2) (x² - 4)/(x - 2) Factor the numerator: x² - 4 = (x - 2)(x + 2) Rewrite the limit: lim(x→2) [(x - 2)(x + 2)]/(x - 2) Cancel the common factor (x - 2): lim(x→2) (x + 2) Substitute x = 2: 2 + 2 = 4 The limit equals 4
   Full step-by-step solution

   Step 1: The original limit is lim(x→2) (x² - 4)/(x - 2) Step 2: Factor the numerator: x² - 4 = (x - 2)(x + 2) Step 3: Rewrite the limit: lim(x→2) [(x - 2)(x + 2)]/(x - 2) Step 4: Cancel the common factor (x - 2): lim(x→2) (x + 2) Step 5: Substitute x = 2: 2 + 2 = 4 Step 6: The limit equals 4

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^(-0.5t) meters, where t is time in years. As time approaches infinity, what height does the tank's capacity approach, and what percentage of the original capacity remains usable? Answer: 1.9 meters, 95% Solution: When analyzing functions with exponential decay terms as the variable approaches infinity, the exponential component approaches zero. This leaves only the constant term, which represents the limiting value.
   Full step-by-step solution

   When analyzing functions with exponential decay terms as the variable approaches infinity, the exponential component approaches zero. This leaves only the constant term, which represents the limiting value. For percentage calculations in proportional containers, the remaining capacity is directly proportional to the remaining height when the cross-sectional area remains constant.

7. lim(x→0) (e^(2x) - 1)/(3x) = ? Answer: 2/3 Solution: Recognize this is a 0/0 indeterminate form since e^(2*0) - 1 = 1 - 1 = 0 and 3*0 = 0. Recall the limit definition: lim(x→0) (e^(ax) - 1)/x = a. Apply this rule with a = 2: lim(x→0) (e^(2x) - 1)/x = 2.
   Full step-by-step solution

   Step 1: Recognize this is a 0/0 indeterminate form since e^(2*0) - 1 = 1 - 1 = 0 and 3*0 = 0. Step 2: Recall the limit definition: lim(x→0) (e^(ax) - 1)/x = a. Step 3: Apply this rule with a = 2: lim(x→0) (e^(2x) - 1)/x = 2. Step 4: Our expression has 3x in the denominator, so we can write: lim(x→0) (e^(2x) - 1)/(3x) = (1/3) × lim(x→0) (e^(2x) - 1)/x. Step 5: Substitute the known limit: (1/3) × 2 = 2/3. The answer is 2/3.