Limits Concept

lessonbunny.com

Answer Key & Explanations

Limits Concept · Grade 12 · Worksheet 3

1. Noah is analyzing the behavior of a function f(x) defined as follows: f(x) = (x^3 - 27)/(x - 3) for x ≠ 3. He creates a table of values for x approaching 3 from both sides. Using the table below, what is the limit of f(x) as x approaches 3? x | 2.9 | 2.99 | 2.999 | 3.001 | 3.01 | 3.1 f(x) | 25.21 | 26.8201 | 26.982001 | 27.018001 | 27.1801 | 28.81 Answer: 27 Solution: Examine the values as x approaches 3 from the left: x = 2.9 gives f(x) = 25.21, x = 2.99 gives f(x) = 26.8201, x = 2.999 gives f(x) = 26.982001. These values are increasing and approaching 27.
   Full step-by-step solution

   Step 1: Examine the values as x approaches 3 from the left: x = 2.9 gives f(x) = 25.21, x = 2.99 gives f(x) = 26.8201, x = 2.999 gives f(x) = 26.982001. These values are increasing and approaching 27. Step 2: Examine the values as x approaches 3 from the right: x = 3.001 gives f(x) = 27.018001, x = 3.01 gives f(x) = 27.1801, x = 3.1 gives f(x) = 28.81. These values are decreasing and approaching 27 from above. Step 3: Since the values from both sides are approaching the same number, 27, the limit exists. Step 4: Therefore, lim(x→3) f(x) = 27. The answer is 27.

2. 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.

3. Olivia is studying the behavior of a function f(x) defined by the following table. As x approaches 3 from both sides, what is the limit of f(x)? x f(x) 2.5 1.75 2.9 1.95 2.99 1.99 2.999 1.999 3.001 2.001 3.01 2.01 3.1 2.1 3.5 2.5 Answer: 2 Solution: Observe the values of f(x) as x approaches 3 from the left (x < 3). For x = 2.5, f(x) = 1.75; for x = 2.9, f(x) = 1.95; for x = 2.99, f(x) = 1.99; for x = 2.999, f(x) = 1.999.
   Full step-by-step solution

   Step 1: Observe the values of f(x) as x approaches 3 from the left (x < 3). For x = 2.5, f(x) = 1.75; for x = 2.9, f(x) = 1.95; for x = 2.99, f(x) = 1.99; for x = 2.999, f(x) = 1.999. As x gets closer to 3 from the left, f(x) approaches 2. Step 2: Observe the values of f(x) as x approaches 3 from the right (x > 3). For x = 3.001, f(x) = 2.001; for x = 3.01, f(x) = 2.01; for x = 3.1, f(x) = 2.1; for x = 3.5, f(x) = 2.5. As x gets closer to 3 from the right, f(x) also approaches 2. Step 3: Since the values of f(x) approach 2 from both the left and the right as x approaches 3, the limit exists and is equal to 2. The answer is 2.

4. Liam is designing a cylindrical water tank for a new apartment building. The tank's volume is given by V(h) = π(25h - h²) cubic meters, where h is the water depth in meters. As the water level approaches 10 meters from below, what value does the volume approach? Answer: 150π Solution: V(h) = π(25h - h²) We want the volume as h approaches 10 meters from below. That means we simply substitute h = 10 into the expression, because V(h) is a polynomial and continuous at h = 10.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** We are given the volume function: V(h) = π(25h - h²) We want the volume as h approaches 10 meters from below. That means we simply substitute h = 10 into the expression, because V(h) is a polynomial and continuous at h = 10. --- **Step 2: Substitute h = 10 into the formula** V(10) = π(25 * 10 - 10²) First compute inside the parentheses: 25 * 10 = 250 10² = 100 So 250 - 100 = 150 Thus: V(10) = π * 150 --- **Step 3: Final answer** V(10) = 150π Since the function is continuous, the limit as h approaches 10 from below is the same as the value at h = 10. --- **Final Answer:** 150π

5. 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.

6. Emma is analyzing the rate at which a chemical reaction occurs in her laboratory experiment. The reaction rate R(x) is modeled by the function R(x) = (x^2 - 16)/(x - 4), where x represents the concentration of a reactant in moles per liter. When examining the behavior as the concentration approaches 4 moles per liter, she notices the function appears undefined at exactly x = 4. What value does the reaction rate approach as the concentration gets closer and closer to 4 moles per liter? Answer: 8 Solution: Start with the function R(x) = (x^2 - 16)/(x - 4) Factor the numerator: x^2 - 16 = (x - 4)(x + 4) Rewrite the function: R(x) = [(x - 4)(x + 4)]/(x - 4) Cancel the common factor (x - 4) from numerator and denominator: R(x) = x + 4 (for x ≠ 4) Find the limit as x approaches 4: lim(x→4) (x + 4) = 4…
   Full step-by-step solution

   Step 1: Start with the function R(x) = (x^2 - 16)/(x - 4) Step 2: Factor the numerator: x^2 - 16 = (x - 4)(x + 4) Step 3: Rewrite the function: R(x) = [(x - 4)(x + 4)]/(x - 4) Step 4: Cancel the common factor (x - 4) from numerator and denominator: R(x) = x + 4 (for x ≠ 4) Step 5: Find the limit as x approaches 4: lim(x→4) (x + 4) = 4 + 4 = 8 Step 6: Therefore, the reaction rate approaches 8 as the concentration approaches 4 moles per liter. The answer is 8.

7. Liam is designing a cylindrical water tank for a new building. The tank's volume is given by V(h) = 50πh cubic meters, where h is the height in meters. Liam needs to determine how the volume changes as the height approaches 4 meters from both above and below. What is the limit of V(h) as h approaches 4? Answer: 200π Solution: We are given the volume function: V(h) = 50πh. We want the limit of V(h) as h approaches 4. Write down the limit expression.
   Full step-by-step solution

   We are given the volume function: V(h) = 50πh. We want the limit of V(h) as h approaches 4. Step 1: Write down the limit expression. We are finding: limit as h -> 4 of V(h) = limit as h -> 4 of (50πh). Step 2: Recognize that V(h) = 50πh is a linear function in h. For a linear function (or any continuous function), the limit as h approaches a value is simply the function evaluated at that value. Step 3: Substitute h = 4 into the function. V(4) = 50π * 4. Step 4: Perform the multiplication. 50 * 4 = 200, so V(4) = 200π. Step 5: State the conclusion. Since the function is continuous at h = 4, the limit from both above and below is the same: 200π. Final answer: 200π