Limits Concept

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Answer Key & Explanations

Limits Concept · Grade 12 · Worksheet 1

1. From graph: as x→7, f(x)→? Answer: 5 Solution: Look at the graph and find the point where x approaches 7. Check the y-values as x approaches 7 from the left (x = 6.9, 6.99, 6.999). The y-values get closer to 5.
   Full step-by-step solution

   Step 1: Look at the graph and find the point where x approaches 7. Step 2: Check the y-values as x approaches 7 from the left (x = 6.9, 6.99, 6.999). The y-values get closer to 5. Step 3: Check the y-values as x approaches 7 from the right (x = 7.1, 7.01, 7.001). The y-values also get closer to 5. Step 4: Since the y-values approach 5 from both sides, the limit is 5. The answer is 5.

2. A particle moves along a curve with its position function given by f(x) = (x^3 - 8)/(x^2 - 4) for x ≠ 2. Determine the limit of f(x) as x approaches 2, which represents the instantaneous position the particle would approach at that moment. 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) Now…
   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) Step 5: Now take 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.

3. lim(x→∞) (3x² + 2x - 1)/(2x² - x + 5) = ? Answer: 3/2 Solution: To find the limit as x approaches infinity of (3x² + 2x - 1)/(2x² - x + 5), we follow these steps: Identify the highest power of x in the denominator. The denominator is 2x² - x + 5. The highest power of x here is x².
   Full step-by-step solution

   To find the limit as x approaches infinity of (3x² + 2x - 1)/(2x² - x + 5), we follow these steps: Step 1: Identify the highest power of x in the denominator. The denominator is 2x² - x + 5. The highest power of x here is x². Step 2: Divide every term in the numerator and the denominator by x². This is the standard method for limits at infinity of rational functions (polynomial divided by polynomial). We divide each term by x². Numerator: (3x²)/x² + (2x)/x² - (1)/x² = 3 + 2/x - 1/x² Denominator: (2x²)/x² - (x)/x² + (5)/x² = 2 - 1/x + 5/x² So the function becomes: (3 + 2/x - 1/x²) / (2 - 1/x + 5/x²) Step 3: Take the limit as x approaches infinity. As x becomes very large (approaches infinity), any term with x in the denominator approaches zero. Specifically: 2/x approaches 0 1/x² approaches 0 1/x approaches 0 5/x² approaches 0 Step 4: Substitute these limiting values into the expression. The expression becomes: (3 + 0 - 0) / (2 - 0 + 0) = 3/2 Step 5: State the final answer. Therefore, the limit is 3/2. This makes sense because for large values of x, the highest degree terms dominate the behavior of the polynomials. The ratio of the leading coefficients (3 from numerator and 2 from denominator) gives us the limit.

4. From graph: as x→10, f(x)→? Answer: 7 Solution: Look at the graph of f(x) as x approaches 10 from the left side As x gets closer to 10 from values like 9.5, 9.8, 9.9, the y-values approach 7 Look at the graph of f(x) as x approaches 10 from the right side As x gets closer to 10 from values like 10.5, 10.2, 10.1, the y-values also approach 7…
   Full step-by-step solution

   Step 1: Look at the graph of f(x) as x approaches 10 from the left side Step 2: As x gets closer to 10 from values like 9.5, 9.8, 9.9, the y-values approach 7 Step 3: Look at the graph of f(x) as x approaches 10 from the right side Step 4: As x gets closer to 10 from values like 10.5, 10.2, 10.1, the y-values also approach 7 Step 5: Since both left-hand and right-hand limits approach the same value of 7, the limit exists Step 6: Therefore, lim(x→10) f(x) = 7

5. From the table: x: 3.9, 3.99, 3.999, 4.001, 4.01, 4.1 f(x): 7.8, 7.98, 7.998, 8.002, 8.02, 8.2 lim(x→4) f(x) = ? Answer: 8 Solution: Examine the f(x) values as x approaches 4 from the left (x = 3.9, 3.99, 3.999).
   Full step-by-step solution

   Step 1: Examine the f(x) values as x approaches 4 from the left (x = 3.9, 3.99, 3.999). - When x = 3.9, f(x) = 7.8 - When x = 3.99, f(x) = 7.98 - When x = 3.999, f(x) = 7.998 As x gets closer to 4 from the left, f(x) gets closer to 8. Step 2: Examine the f(x) values as x approaches 4 from the right (x = 4.001, 4.01, 4.1). - When x = 4.001, f(x) = 8.002 - When x = 4.01, f(x) = 8.02 - When x = 4.1, f(x) = 8.2 As x gets closer to 4 from the right, f(x) also gets closer to 8. Step 3: Since the left-hand limit (as x→4⁻) and the right-hand limit (as x→4⁺) both approach 8, the limit exists and is equal to 8. The answer is 8.

6. A pharmaceutical company is modeling the concentration of a new medication in the bloodstream over time. The concentration function is given by C(t) = (3t^2 - 12) / (t^2 - 4) for t > 2 hours. As time continues indefinitely, what value does the drug concentration approach in the bloodstream? Answer: 3 Solution: C(t) = (3t^2 - 12) / (t^2 - 4) for t > 2. As t becomes very large, both the numerator and denominator become dominated by the highest power of t, which is t^2.
   Full step-by-step solution

   Let's solve this step by step. We are given the concentration function: C(t) = (3t^2 - 12) / (t^2 - 4) for t > 2. --- **Step 1: Identify what happens as t → ∞** As t becomes very large, both the numerator and denominator become dominated by the highest power of t, which is t^2. --- **Step 2: Factor numerator and denominator** Factor numerator: 3t^2 - 12 = 3(t^2 - 4) Factor denominator: t^2 - 4 = (t - 2)(t + 2) but we don't need the full factoring for the limit. Actually, notice: C(t) = (3t^2 - 12) / (t^2 - 4) = 3(t^2 - 4) / (t^2 - 4) --- **Step 3: Simplify the function** For t > 2, t^2 - 4 ≠ 0, so we can cancel: C(t) = 3(t^2 - 4) / (t^2 - 4) = 3 --- **Step 4: Interpret the result** The function simplifies exactly to C(t) = 3 for all t > 2 (except at points where denominator is zero, but t > 2 avoids t = 2). So as t → ∞, C(t) = 3 exactly. --- **Step 5: Conclusion** The concentration approaches 3 mg/L (or whatever units) as time continues indefinitely. --- **Final answer:** 3

7. 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.

8. 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.