Limit Calculation

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Limit Calculation · Grade 12 · Worksheet 2

1. A biomedical engineer is modeling the concentration of a new drug in a patient's bloodstream over time. The concentration function is given by C(t) = (t² - 4)/(t - 2) for t ≠ 2, where t is measured in hours. To understand the drug's behavior at the critical time t = 2 hours, the engineer needs to determine the limit of C(t) as t approaches 2. What is this limit value? Answer: 4 Solution: Write down the function. C(t) = (t² - 4)/(t - 2) for t ≠ 2. Direct substitution.
   Full step-by-step solution

   Let's find the limit of C(t) as t approaches 2. Step 1: Write down the function. C(t) = (t² - 4)/(t - 2) for t ≠ 2. Step 2: Direct substitution. If we try to plug in t = 2 directly, we get: Numerator: (2)² - 4 = 4 - 4 = 0 Denominator: 2 - 2 = 0 This gives 0/0, which is an indeterminate form. This means we need to simplify the function. Step 3: Factor the numerator. Notice that t² - 4 is a difference of squares. t² - 4 = (t - 2)(t + 2) Step 4: Rewrite the function. C(t) = (t - 2)(t + 2)/(t - 2) Step 5: Simplify the function. Since t ≠ 2, the term (t - 2) in the numerator and denominator cancels out. C(t) = t + 2, for t ≠ 2. Step 6: Take the limit. The limit as t approaches 2 of C(t) is now the limit as t approaches 2 of (t + 2). lim (t→2) (t + 2) = 2 + 2 = 4. Step 7: Conclusion. Even though the original function is undefined at t = 2, the values of C(t) get arbitrarily close to 4 as t gets close to 2. Therefore, the limit is 4. ANSWER: 4

2. A biomedical engineer is modeling the concentration of a new drug in a patient's bloodstream over time. The concentration function is given by C(t) = (3t² - 12)/(t² - 4) milligrams per liter, where t is time in hours after administration. As time approaches 2 hours, the function appears undefined. Determine what the concentration approaches as t gets arbitrarily close to 2 hours from both sides. Answer: 3 Solution: C(t) = (3t² - 12) / (t² - 4) and we want the limit as t approaches 2. At t = 2: C(2) = (3*(2)² - 12) / ((2)² - 4) = (12 - 12) / (4 - 4) = 0/0 This is an indeterminate form, so we need to simplify.
   Full step-by-step solution

   Let's solve step by step. We are given: C(t) = (3t² - 12) / (t² - 4) and we want the limit as t approaches 2. --- **Step 1: Direct substitution check** At t = 2: C(2) = (3*(2)² - 12) / ((2)² - 4) = (12 - 12) / (4 - 4) = 0/0 This is an indeterminate form, so we need to simplify. --- **Step 2: Factor numerator and denominator** Numerator: 3t² - 12 = 3(t² - 4) Denominator: t² - 4 = (t - 2)(t + 2) So: C(t) = [3(t² - 4)] / [(t - 2)(t + 2)] But t² - 4 = (t - 2)(t + 2) as well. --- **Step 3: Cancel common factor** C(t) = [3 * (t - 2)(t + 2)] / [(t - 2)(t + 2)] Cancel (t - 2)(t + 2) from numerator and denominator (for t ≠ 2 and t ≠ -2): C(t) = 3 --- **Step 4: Interpret the result** After simplification, C(t) = 3 for all t except t = 2 and t = -2, where it is undefined. Therefore, as t approaches 2 from either side, C(t) approaches 3. --- **Final answer:** 3

3. Evaluate the limit: lim(x→∞) [√(4x² + 3x - 1) - 2x] Answer: 0.75 Solution: limit as x → ∞ of [ sqrt(4x² + 3x - 1) - 2x ] As x → ∞, sqrt(4x² + 3x - 1) ≈ sqrt(4x²) = 2x. So we have 2x - 2x = 0? Not exactly — the 3x - 1 inside the square root matters for a finite limit.
   Full step-by-step solution

   Let's evaluate the limit step by step. We have: limit as x → ∞ of [ sqrt(4x² + 3x - 1) - 2x ] --- **Step 1: Recognize the ∞ - ∞ form** As x → ∞, sqrt(4x² + 3x - 1) ≈ sqrt(4x²) = 2x. So we have 2x - 2x = 0? Not exactly — the 3x - 1 inside the square root matters for a finite limit. This is an ∞ - ∞ indeterminate form. --- **Step 2: Multiply and divide by the conjugate** Multiply numerator and denominator by sqrt(4x² + 3x - 1) + 2x: [ sqrt(4x² + 3x - 1) - 2x ] * [ sqrt(4x² + 3x - 1) + 2x ] / [ sqrt(4x² + 3x - 1) + 2x ] --- **Step 3: Simplify numerator** Numerator = (4x² + 3x - 1) - (2x)² = 4x² + 3x - 1 - 4x² = 3x - 1 So the expression becomes: (3x - 1) / [ sqrt(4x² + 3x - 1) + 2x ] --- **Step 4: Factor x from numerator and denominator** Factor x from numerator: x(3 - 1/x) Factor x from denominator: sqrt(x²(4 + 3/x - 1/x²)) + 2x = x * sqrt(4 + 3/x - 1/x²) + 2x = x [ sqrt(4 + 3/x - 1/x²) + 2 ] So the expression becomes: [ x(3 - 1/x) ] / [ x ( sqrt(4 + 3/x - 1/x²) + 2 ) ] Cancel x (x > 0): = (3 - 1/x) / [ sqrt(4 + 3/x - 1/x²) + 2 ] --- **Step 5: Take the limit as x → ∞** As x → ∞: 1/x → 0, 1/x² → 0 So 3 - 1/x → 3 sqrt(4 + 3/x - 1/x²) → sqrt(4) = 2 Denominator → 2 + 2 = 4 Thus the limit = 3 / 4 = 0.75 --- **Final Answer:** 0.75

4. Consider the curve described by the equation y = (x² - 25) / (x - 5) for x ≠ 5. On a coordinate plane, this curve is a straight line with a hole at x = 5. Hana and Matiu are discussing what happens as x approaches 5 from both sides. They notice the graph approaches a single point, but the function is undefined at x = 5. Find the limit of y as x approaches 5, using algebraic manipulation. Answer: 10 Solution: Write the limit expression: lim(x→5) (x² - 25) / (x - 5). Factor the numerator: x² - 25 = (x - 5)(x + 5). Rewrite the expression: ( (x - 5)(x + 5) ) / (x - 5).
   Full step-by-step solution

   Step 1: Write the limit expression: lim(x→5) (x² - 25) / (x - 5). Step 2: Factor the numerator: x² - 25 = (x - 5)(x + 5). Step 3: Rewrite the expression: ( (x - 5)(x + 5) ) / (x - 5). Step 4: Cancel the common factor (x - 5), provided x ≠ 5: the expression simplifies to x + 5. Step 5: Now evaluate the limit: lim(x→5) (x + 5) = 5 + 5 = 10. The answer is 10.

5. 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.

6. 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.

7. A geometric pattern is formed by constructing a sequence of right triangles on a coordinate plane. The first triangle has vertices at (0,0), (1,0), and (0,1). Each subsequent triangle is constructed by taking the hypotenuse of the previous triangle as one leg and drawing a perpendicular leg of equal length. If this process continues infinitely, what is the limit of the x-coordinate of the point where the hypotenuses of all triangles would intersect if extended? Answer: 2/3 Solution: Identify the pattern of hypotenuse endpoints. The first triangle has hypotenuse endpoints (0,0) and (1,1).
   Full step-by-step solution

   Step 1: Identify the pattern of hypotenuse endpoints. The first triangle has hypotenuse endpoints (0,0) and (1,1). Step 2: The second triangle uses the hypotenuse from (0,0) to (1,1) as one leg, and adds a perpendicular leg of equal length. This creates a new hypotenuse from (0,0) to (2,1). Step 3: The third triangle uses the hypotenuse from (0,0) to (2,1) as one leg, and adds a perpendicular leg of equal length. This creates a new hypotenuse from (0,0) to (3,2). Step 4: The fourth triangle creates a hypotenuse from (0,0) to (5,3). Step 5: The sequence of hypotenuse endpoints is: (1,1), (2,1), (3,2), (5,3), (8,5), (13,8)... Step 6: The x-coordinates follow the Fibonacci sequence: 1, 2, 3, 5, 8, 13... Step 7: The ratio of consecutive Fibonacci numbers approaches the golden ratio φ = (1+sqrt(5))/2 ≈ 1.618. Step 8: However, we need the limit point where all extended hypotenuses would intersect. This occurs where the line y = (1/2)x + 1/3 intersects y = x. Step 9: Set x = (1/2)x + 1/3 Step 10: x - (1/2)x = 1/3 Step 11: (1/2)x = 1/3 Step 12: x = 2/3 The answer is 2/3.