Limit Calculation

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Answer Key & Explanations

Limit Calculation · Grade 12 · Worksheet 2

1. A geometric pattern is formed by a sequence of squares where the side length of each square is half the previous one. The first square has a side length of 8 cm. What is the total area of all the squares in the pattern as the number of squares approaches infinity? Answer: 85.33 Solution: We have a sequence of squares. The first square has side length \( 8 \) cm. Each subsequent square has side length half of the previous one.
   Full step-by-step solution

   Let's solve this step by step. --- **Step 1: Understand the problem** We have a sequence of squares. The first square has side length \( 8 \) cm. Each subsequent square has side length half of the previous one. We want the total area of all squares as the number of squares → ∞. --- **Step 2: Find the area of the first square** Area of first square = side × side = \( 8 \times 8 = 64 \) cm². --- **Step 3: Find the area of the second square** Side length of second square = \( 8 \times (1/2) = 4 \) cm. Area = \( 4 \times 4 = 16 \) cm². --- **Step 4: Find the area of the third square** Side length of third square = \( 4 \times (1/2) = 2 \) cm. Area = \( 2 \times 2 = 4 \) cm². --- **Step 5: Recognize the pattern in areas** Areas: 64, 16, 4, 1, 0.25, … Check ratio: 16 / 64 = 1/4 4 / 16 = 1/4 So each area is 1/4 of the previous area. --- **Step 6: Set up the infinite geometric series** First term \( a = 64 \) Common ratio \( r = 1/4 \) Sum of infinite geometric series = \( a / (1 - r) \), provided \( |r| < 1 \). --- **Step 7: Apply the formula** Sum = \( 64 / (1 - 1/4) \) = \( 64 / (3/4) \) = \( 64 \times (4/3) \) = \( 256 / 3 \). --- **Step 8: Convert to decimal** 256 / 3 = 85.333… Rounded to two decimal places: 85.33 --- **Final Answer:** 85.33

2. A pharmaceutical company is modeling the concentration of a new drug in a patient's bloodstream over time. The concentration function is given by C(t) = (3t² - 5t - 2)/(t² - 4) milligrams per liter, where t is time in hours. As time approaches 2 hours, the company needs to determine what concentration the drug is approaching in the bloodstream. What is the limit of C(t) as t approaches 2? Answer: 7/4 Solution: This means we need to simplify the function. Factor the numerator: 3t² - 5t - 2 We look for factors of (3)(-2) = -6 that add to -5. These are -6 and +1.
   Full step-by-step solution

   Let's find the limit of C(t) as t approaches 2, where: C(t) = (3t² - 5t - 2) / (t² - 4) Step 1: Direct substitution Substitute t = 2 into the function: Numerator: 3(2)² - 5(2) - 2 = 3(4) - 10 - 2 = 12 - 10 - 2 = 0 Denominator: (2)² - 4 = 4 - 4 = 0 We get 0/0, which is indeterminate. This means we need to simplify the function. Step 2: Factor both numerator and denominator Factor the numerator: 3t² - 5t - 2 We look for factors of (3)(-2) = -6 that add to -5. These are -6 and +1. Rewrite: 3t² - 6t + t - 2 Group: (3t² - 6t) + (t - 2) = 3t(t - 2) + 1(t - 2) = (t - 2)(3t + 1) Factor the denominator: t² - 4 This is a difference of squares: (t - 2)(t + 2) Step 3: Simplify the function C(t) = [(t - 2)(3t + 1)] / [(t - 2)(t + 2)] Since t is approaching 2 but not equal to 2, we can cancel the common factor (t - 2): C(t) = (3t + 1) / (t + 2) Step 4: Evaluate the limit using the simplified form Now take the limit as t approaches 2 of the simplified function: lim(t→2) (3t + 1)/(t + 2) = (3(2) + 1)/(2 + 2) = (6 + 1)/4 = 7/4 Therefore, as time approaches 2 hours, the drug concentration approaches 7/4 milligrams per liter. Final answer: 7/4

3. Evaluate the limit: lim(x→0) (e^(2x) - 1 - 2x) / (x^2) Answer: 2 Solution: As x→0, e^(2x)→1, so numerator→1-1-0=0 and denominator→0. This is a 0/0 indeterminate form. Apply L'Hôpital's rule.
   Full step-by-step solution

   Step 1: First, verify that this is an indeterminate form. As x→0, e^(2x)→1, so numerator→1-1-0=0 and denominator→0. This is a 0/0 indeterminate form. Step 2: Apply L'Hôpital's rule. Differentiate numerator and denominator separately. Derivative of numerator: d/dx[e^(2x) - 1 - 2x] = 2e^(2x) - 2 Derivative of denominator: d/dx[x^2] = 2x Step 3: The limit becomes: lim(x→0) (2e^(2x) - 2)/(2x) Step 4: This is still 0/0 indeterminate form, so apply L'Hôpital's rule again. Derivative of numerator: d/dx[2e^(2x) - 2] = 4e^(2x) Derivative of denominator: d/dx[2x] = 2 Step 5: The limit becomes: lim(x→0) 4e^(2x)/2 = lim(x→0) 2e^(2x) Step 6: Evaluate at x=0: 2e^(2×0) = 2e^0 = 2×1 = 2 The answer is 2.

4. A civil engineer is designing a suspension bridge where the cable shape follows the function f(x) = (x^3 - 8)/(x^2 - 4) for x ≠ 2. To ensure structural integrity at the critical support point x = 2, she needs to determine what height the cable approaches as it gets closer to this point. What is the limit of f(x) as x approaches 2? Answer: 3 Solution: Write the original function: f(x) = (x^3 - 8)/(x^2 - 4) Factor the numerator using difference of cubes: x^3 - 8 = (x - 2)(x^2 + 2x + 4) Factor the denominator using difference of squares: x^2 - 4 = (x - 2)(x + 2) Simplify by canceling the common factor (x - 2): f(x) = (x^2 + 2x + 4)/(x + 2) for…
   Full step-by-step solution

   Step 1: Write the original function: f(x) = (x^3 - 8)/(x^2 - 4) Step 2: Factor the numerator using difference of cubes: x^3 - 8 = (x - 2)(x^2 + 2x + 4) Step 3: Factor the denominator using difference of squares: x^2 - 4 = (x - 2)(x + 2) Step 4: Simplify by canceling the common factor (x - 2): f(x) = (x^2 + 2x + 4)/(x + 2) for x ≠ 2 Step 5: Evaluate the limit as x approaches 2: lim(x→2) (x^2 + 2x + 4)/(x + 2) Step 6: Substitute x = 2 into the simplified expression: (2^2 + 2(2) + 4)/(2 + 2) = (4 + 4 + 4)/4 = 12/4 = 3 The answer is 3.

5. lim_(x→∞) (4x⁴ - 7x² + 9)/(2x⁴ + 5x³ - 3) = ? Answer: 2 Solution: Identify the highest power of x in both numerator and denominator. Both are degree 4. Divide every term in numerator and denominator by x⁴.
   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 in numerator and denominator by x⁴. Step 3: The expression becomes (4 - 7/x² + 9/x⁴)/(2 + 5/x - 3/x⁴). Step 4: As x approaches infinity, all 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. lim_(x→∞) (5x³ - 2x² + 7)/(3x³ + 4x - 1) = ? Answer: 5/3 Solution: Identify the degrees of the numerator and denominator. Both are degree 3 polynomials.
   Full step-by-step solution

   Step 1: Identify the degrees of the numerator and denominator. Both are degree 3 polynomials. Step 2: For rational functions where the numerator and denominator have the same degree, the limit as x approaches infinity equals the ratio of the leading coefficients. Step 3: The leading coefficient of the numerator is 5, and the leading coefficient of the denominator is 3. Step 4: Therefore, the limit equals 5/3. The answer is 5/3.