Limits Concept

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Limits Concept · Grade 12 · Worksheet 2

1. Liam is designing a cylindrical water tank for a new building. The tank's volume must be exactly 1000 cubic meters. To minimize material costs, he needs to find the dimensions that minimize the surface area. If the tank has a circular base with radius r meters and height h meters, what ratio h/r minimizes the surface area while maintaining the fixed volume? Answer: 2 Solution: Write down the known formulas. V = π r² h Given: V = 1000 m³ π r² h = 1000 … (1) Surface area of a closed cylinder (top, bottom, and side): A = 2πr² + 2πrh … (2) Express h in terms of r using the volume constraint.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Write down the known formulas.** Volume of a cylinder: V = π r² h Given: V = 1000 m³ So: π r² h = 1000 … (1) Surface area of a closed cylinder (top, bottom, and side): A = 2πr² + 2πrh … (2) --- **Step 2: Express h in terms of r using the volume constraint.** From (1): h = 1000 / (π r²) --- **Step 3: Substitute h into the surface area formula.** A(r) = 2πr² + 2πr * [1000 / (π r²)] Simplify: A(r) = 2πr² + (2000 / r) --- **Step 4: Minimize A(r) with respect to r.** Differentiate A(r) with respect to r: A'(r) = 4πr - 2000 / r² Set A'(r) = 0: 4πr - 2000 / r² = 0 4πr = 2000 / r² Multiply both sides by r²: 4πr³ = 2000 r³ = 2000 / (4π) r³ = 500 / π So: r = (500 / π)^(1/3) --- **Step 5: Find h corresponding to this r.** From h = 1000 / (π r²): h = 1000 / [ π * (500/π)^(2/3) ] Simplify: h = 1000 / [ π * (500^(2/3) / π^(2/3)) ] h = 1000 / [ π^(1 - 2/3) * 500^(2/3) ] h = 1000 / [ π^(1/3) * 500^(2/3) ] --- **Step 6: Compute the ratio h / r.** h / r = { 1000 / [ π^(1/3) * 500^(2/3) ] } / { (500/π)^(1/3) } Simplify denominator r: r = (500/π)^(1/3) = 500^(1/3) / π^(1/3) So: h / r = [ 1000 / (π^(1/3) * 500^(2/3)) ] * [ π^(1/3) / 500^(1/3) ] π^(1/3) cancels: h / r = 1000 / [ 500^(2/3) * 500^(1/3) ] h / r = 1000 / 500^(1) h / r = 1000 / 500 h / r = 2 --- **Step 7: Conclusion.** The ratio h / r that minimizes the surface area for a fixed volume is 2. --- **Final answer:** 2

2. From the table: x: 1.97, 1.98, 1.99, 2.00, 2.01, 2.02, 2.03 f(x): 7.88, 7.92, 7.96, 8.00, 8.04, 8.08, 8.12 lim(x→2) f(x) = ? Answer: 8 Solution: Examine the f(x) values as x approaches 2 from the left (x = 1.97, 1.98, 1.99). The corresponding f(x) values are 7.88, 7.92, 7.96. These values are getting closer to 8.
   Full step-by-step solution

   Step 1: Examine the f(x) values as x approaches 2 from the left (x = 1.97, 1.98, 1.99). The corresponding f(x) values are 7.88, 7.92, 7.96. These values are getting closer to 8. Step 2: Examine the f(x) values as x approaches 2 from the right (x = 2.01, 2.02, 2.03). The corresponding f(x) values are 8.04, 8.08, 8.12. These values are also getting closer to 8 from above. Step 3: At x = 2.00, f(x) = 8.00. Step 4: Since the f(x) values approach 8 from both sides, the limit as x approaches 2 is 8. The answer is 8.

3. Emma is analyzing the temperature change in a chemical reaction. The temperature T(t) in degrees Celsius after t minutes is modeled by the function T(t) = (t^2 - 16)/(t - 4). When examining the temperature behavior as time approaches 4 minutes, she notices the function appears undefined at exactly t=4. What temperature does the reaction approach as time gets closer and closer to 4 minutes? Answer: 8 Solution: The function is T(t) = (t^2 - 16)/(t - 4) Factor the numerator: t^2 - 16 = (t - 4)(t + 4) Simplify the expression: T(t) = [(t - 4)(t + 4)]/(t - 4) = t + 4 (for t ≠ 4) As t approaches 4, T(t) approaches 4 + 4 = 8 The answer is 8.
   Full step-by-step solution

   Step 1: The function is T(t) = (t^2 - 16)/(t - 4) Step 2: Factor the numerator: t^2 - 16 = (t - 4)(t + 4) Step 3: Simplify the expression: T(t) = [(t - 4)(t + 4)]/(t - 4) = t + 4 (for t ≠ 4) Step 4: As t approaches 4, T(t) approaches 4 + 4 = 8 The answer is 8.

4. Liam is designing a new smartphone app that tracks user engagement. He models the number of active users over time with the function f(t) = (t² - 9)/(t - 3), where t represents weeks after launch. However, the function appears undefined at t = 3 weeks. Liam needs to determine what value the number of active users approaches as time gets closer to 3 weeks after launch. What is this approaching value? Answer: 6 Solution: f(t) = (t² - 9)/(t - 3) Identify the problem at t = 3 If we plug in t = 3 directly: Numerator: (3² - 9) = (9 - 9) = 0 Denominator: (3 - 3) = 0 So f(3) = 0/0, which is undefined.
   Full step-by-step solution

   Let's work through this step-by-step. We have the function: f(t) = (t² - 9)/(t - 3) --- **Step 1: Identify the problem at t = 3** If we plug in t = 3 directly: Numerator: (3² - 9) = (9 - 9) = 0 Denominator: (3 - 3) = 0 So f(3) = 0/0, which is undefined. --- **Step 2: Factor the numerator** Notice that t² - 9 is a difference of squares: t² - 9 = (t - 3)(t + 3) --- **Step 3: Simplify the function** f(t) = [(t - 3)(t + 3)] / (t - 3) For t ≠ 3, we can cancel (t - 3) from numerator and denominator: f(t) = t + 3 (for t ≠ 3) --- **Step 4: Find the limit as t approaches 3** Even though the original function is undefined at t = 3, the simplified function f(t) = t + 3 is defined and continuous everywhere. So as t → 3, f(t) → 3 + 3 = 6. --- **Step 5: Conclusion** The number of active users approaches 6 as t gets closer to 3 weeks. --- **Final answer:** 6

5. From the table: x: 3.7, 3.9, 3.99, 4.01, 4.1, 4.3 f(x): 11.1, 11.7, 11.97, 12.03, 12.3, 12.9 lim(x→4) f(x) = ? Answer: 12 Solution: Examine the left-hand limit as x approaches 4 from below. The x values are 3.7, 3.9, 3.99 with corresponding f(x) values 11.1, 11.7, 11.97. As x gets closer to 4 from the left, f(x) approaches 12.
   Full step-by-step solution

   Step 1: Examine the left-hand limit as x approaches 4 from below. The x values are 3.7, 3.9, 3.99 with corresponding f(x) values 11.1, 11.7, 11.97. As x gets closer to 4 from the left, f(x) approaches 12. Step 2: Examine the right-hand limit as x approaches 4 from above. The x values are 4.01, 4.1, 4.3 with corresponding f(x) values 12.03, 12.3, 12.9. As x gets closer to 4 from the right, f(x) also approaches 12. Step 3: Since both the left-hand limit and the right-hand limit approach the same value of 12, the limit exists. Step 4: Therefore, lim(x→4) f(x) = 12. The answer is 12.

6. From the table: x: 1.97, 1.98, 1.99, 2.00, 2.01, 2.02, 2.03 f(x): 3.94, 3.96, 3.98, 4.00, 4.02, 4.04, 4.06 lim(x→2) f(x) = ? Answer: 4 Solution: Examine the table for x values approaching 2 from the left: x = 1.97, 1.98, 1.99. The corresponding f(x) values are 3.94, 3.96, 3.98. As x gets closer to 2 from the left, f(x) approaches 4.
   Full step-by-step solution

   Step 1: Examine the table for x values approaching 2 from the left: x = 1.97, 1.98, 1.99. The corresponding f(x) values are 3.94, 3.96, 3.98. As x gets closer to 2 from the left, f(x) approaches 4. Step 2: Examine the table for x values approaching 2 from the right: x = 2.01, 2.02, 2.03. The corresponding f(x) values are 4.02, 4.04, 4.06. As x gets closer to 2 from the right, f(x) also approaches 4. Step 3: At x = 2.00, f(x) = 4.00, confirming the trend. Step 4: Since the left-hand limit and right-hand limit both approach 4, the limit exists and equals 4. The answer is 4.

7. lim(x→0) (sin(3x))/(2x) = ? Answer: 1.5 Solution: Recall the standard limit: lim(θ→0) sin(θ)/θ = 1 We have lim(x→0) sin(3x)/(2x) Multiply numerator and denominator by 3/3: (3/3) × sin(3x)/(2x) = (3 sin(3x))/(6x) Rewrite as (3/2) × (sin(3x))/(3x) As x→0, 3x→0, so (sin(3x))/(3x) → 1 Therefore, the limit equals (3/2) × 1 = 3/2 = 1.5 The answer is 1.5.
   Full step-by-step solution

   Step 1: Recall the standard limit: lim(θ→0) sin(θ)/θ = 1 Step 2: We have lim(x→0) sin(3x)/(2x) Step 3: Multiply numerator and denominator by 3/3: (3/3) × sin(3x)/(2x) = (3 sin(3x))/(6x) Step 4: Rewrite as (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 = 1.5 The answer is 1.5.

8. 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