Limits Concept

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Answer Key & Explanations

Limits Concept · Grade 12 · Worksheet 1

1. Aroha is tracking the temperature of a cooling liquid in a physics experiment. She records the temperature T(t) in degrees Celsius at various times t (in minutes) using the function T(t) = (t^3 - 27)/(t - 3). The function is undefined at exactly t = 3 minutes. Using the table of values below, determine what temperature the liquid approaches as time gets closer and closer to 3 minutes. t (minutes) | T(t) (°C) 2.9 | 26.11 2.99 | 26.9101 2.999 | 26.991001 3.001 | 27.009001 3.01 | 27.0901 3.1 | 27.91 Answer: 27 Solution: Examine the table values as t approaches 3 from the left (t < 3): - At t = 2.9, T = 26.11 - At t = 2.99, T = 26.9101 - At t = 2.999, T = 26.991001 These values are increasing and getting very close to 27.
   Full step-by-step solution

   Step 1: Examine the table values as t approaches 3 from the left (t < 3): - At t = 2.9, T = 26.11 - At t = 2.99, T = 26.9101 - At t = 2.999, T = 26.991001 These values are increasing and getting very close to 27. Step 2: Examine the table values as t approaches 3 from the right (t > 3): - At t = 3.001, T = 27.009001 - At t = 3.01, T = 27.0901 - At t = 3.1, T = 27.91 These values are decreasing and also getting very close to 27. Step 3: Since both sides approach the same number, 27, the limit exists. The temperature approaches 27 degrees Celsius as time approaches 3 minutes. The answer is 27.

2. lim(x→1) (x² - 1)/(x - 1) = ? Answer: 2 Solution: Factor the numerator: x² - 1 = (x - 1)(x + 1) Rewrite the limit: lim(x→1) [(x - 1)(x + 1)]/(x - 1) Cancel the common factor (x - 1): lim(x→1) (x + 1) Substitute x = 1: 1 + 1 = 2 The answer is 2.
   Full step-by-step solution

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

3. From the table: x: 0.96, 0.98, 0.99, 1.00, 1.01, 1.02, 1.04 f(x): 2.81, 2.91, 2.96, undefined, 3.06, 3.11, 3.21 What is lim(x→1) f(x)? Answer: 3.01 Solution: Examine the left-hand limit as x approaches 1 from below. The x values 0.96, 0.98, 0.99 give f(x) values 2.81, 2.91, 2.96. These are approaching 3.01.
   Full step-by-step solution

   Step 1: Examine the left-hand limit as x approaches 1 from below. The x values 0.96, 0.98, 0.99 give f(x) values 2.81, 2.91, 2.96. These are approaching 3.01. Step 2: Examine the right-hand limit as x approaches 1 from above. The x values 1.01, 1.02, 1.04 give f(x) values 3.06, 3.11, 3.21. These are also approaching 3.01 from above. Step 3: Since both the left-hand and right-hand limits approach the same value, 3.01, the limit exists. Step 4: Therefore, lim(x→1) f(x) = 3.01.

4. Liam is designing a new roller coaster and needs to model the steepness of the initial drop. The height of the track above the ground, in meters, is given by the function h(t) = (t² - 9) / (t - 3) for t ≠ 3, where t is the horizontal distance from the start in meters. To ensure a smooth and continuous track, he needs to determine what height the track should be at exactly t = 3 meters. What value should Liam use for h(3) to fill the gap in his function? Answer: 6 Solution: h(t) = (t² - 9) / (t - 3) for t ≠ 3. We need to find h(3) so the track is continuous at t = 3.
   Full step-by-step solution

   Let's go step-by-step. We are given: h(t) = (t² - 9) / (t - 3) for t ≠ 3. We need to find h(3) so the track is continuous at t = 3. --- **Step 1: Identify the problem at t = 3** If we plug t = 3 into h(t) directly: Numerator: 3² - 9 = 9 - 9 = 0 Denominator: 3 - 3 = 0 So h(3) would be 0/0, which is undefined. This means there is a removable discontinuity (a hole) in the graph at t = 3. --- **Step 2: Simplify the function for t ≠ 3** Factor the numerator: t² - 9 = (t - 3)(t + 3) So h(t) = (t - 3)(t + 3) / (t - 3) For t ≠ 3, we can cancel (t - 3) from numerator and denominator: h(t) = t + 3, for t ≠ 3. --- **Step 3: Find the limit as t approaches 3** After simplification, h(t) = t + 3 for t ≠ 3. So as t → 3, h(t) → 3 + 3 = 6. --- **Step 4: Define h(3) for continuity** To make the track continuous at t = 3, we define h(3) to be the limit: h(3) = 6. --- **Final Answer:** 6

5. lim(x→∞) (4x⁴ - 3x² + 5)/(2x⁴ + x³ - 7) = ? 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/x² + 5/x⁴)/(2 + 1/x - 7/x⁴). Step 4: As x approaches infinity, 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.

6. Matiu is studying the velocity of a drone as it approaches a landing pad. The drone's height above the ground (in meters) at time t seconds is given by the function h(t) = (t^3 - 27) / (t - 3), which is undefined at t = 3 seconds. To program a smooth landing, Matiu needs to know the height the drone approaches as t gets very close to 3 seconds from either side. What value does the height approach? Answer: 27 Solution: Write the function: h(t) = (t^3 - 27) / (t - 3) Recognize that the numerator is a difference of cubes: t^3 - 27 = t^3 - 3^3 Use the difference of cubes formula: a^3 - b^3 = (a - b)(a^2 + ab + b^2).
   Full step-by-step solution

   Step 1: Write the function: h(t) = (t^3 - 27) / (t - 3) Step 2: Recognize that the numerator is a difference of cubes: t^3 - 27 = t^3 - 3^3 Step 3: Use the difference of cubes formula: a^3 - b^3 = (a - b)(a^2 + ab + b^2). Here a = t, b = 3, so t^3 - 27 = (t - 3)(t^2 + 3t + 9) Step 4: Rewrite the function: h(t) = [(t - 3)(t^2 + 3t + 9)] / (t - 3) Step 5: Cancel the common factor (t - 3) for t ≠ 3: h(t) = t^2 + 3t + 9 Step 6: Find the limit as t approaches 3: lim(t→3) (t^2 + 3t + 9) = (3)^2 + 3(3) + 9 = 9 + 9 + 9 = 27 The answer is 27.

7. 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.