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Limit Calculation

Grade 12 · Calculus · Worksheet 1

  1. Aroha sketches the graph of the function f(x) = (x^3 - 64) / (x - 4). On the coordinate plane, the graph appears to be a smooth curve everywhere except at x = 4, where there is a hole. What is the y-coordinate of that hole? Answer: ______________
  2. lim_{x→7} (x² - 49)/(x - 7) = ? Answer: ______________
  3. lim_(x→∞) (4x³ - 7x + 2)/(2x³ + 5x² - 3) = ? Answer: ______________
  4. Aroha is a geologist studying the rate of sediment accumulation in a river delta. The depth of sediment (in meters) is modeled by the function S(t) = (t^3 - 125)/(t - 5) for t ≠ 5, where t represents time in thousands of years since the delta began forming. Due to a core sample gap, the depth at exactly t = 5 thousand years was not recorded. Aroha needs to determine what depth the sediment approaches as time gets arbitrarily close to 5 thousand years from both sides. What is the limit of S(t) as t approaches 5? Answer: ______________
  5. Sophia is analyzing the graph of the function f(x) = (sqrt(2x + 1) - 1) / (x) for x ≠ 0. On a coordinate plane, the graph appears to have a hole at x = 0. As x approaches 0 from both sides, the y-values approach a specific number. Determine the limit of f(x) as x approaches 0 by using algebraic manipulation. Answer: ______________
  6. Hana is an environmental scientist studying the rate of pollutant decay in a lake. The concentration of the pollutant (in parts per million) is modeled by the function C(x) = (4x^2 - 36)/(x - 3) for x ≠ 3, where x represents time in days after a chemical spill. To determine the immediate concentration at the critical time x = 3 days, Hana needs to find what value the concentration approaches as time gets arbitrarily close to 3 days from both sides. What is the limit of C(x) as x approaches 3? Answer: ______________
  7. Olivia sketches the graph of the function f(x) = (x^2 - 10x + 25) / (x^2 - 25). On the coordinate plane, she notices a hole in the graph at x = 5. As x approaches 5 from both sides, what y-value does the function approach? Answer: ______________
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Answer Key & Explanations

Limit Calculation · Grade 12 · Worksheet 1

  1. Aroha sketches the graph of the function f(x) = (x^3 - 64) / (x - 4). On the coordinate plane, the graph appears to be a smooth curve everywhere except at x = 4, where there is a hole. What is the y-coordinate of that hole? Answer: 48 Solution: Write the limit: lim(x→4) (x^3 - 64) / (x - 4). Factor the numerator as difference of cubes: x^3 - 64 = (x - 4)(x^2 + 4x + 16). Rewrite the expression: ((x - 4)(x^2 + 4x + 16)) / (x - 4).
    Full step-by-step solution

    Step 1: Write the limit: lim(x→4) (x^3 - 64) / (x - 4). Step 2: Factor the numerator as difference of cubes: x^3 - 64 = (x - 4)(x^2 + 4x + 16). Step 3: Rewrite the expression: ((x - 4)(x^2 + 4x + 16)) / (x - 4). Step 4: Cancel the common factor (x - 4) (since x ≠ 4): x^2 + 4x + 16. Step 5: Now substitute x = 4: 4^2 + 4(4) + 16 = 16 + 16 + 16 = 48. The answer is 48.

  2. lim_{x→7} (x² - 49)/(x - 7) = ? Answer: 14 Solution: Direct substitution gives (7² - 49)/(7 - 7) = (49 - 49)/0 = 0/0, an indeterminate form. Factor the numerator: x² - 49 = (x - 7)(x + 7). Rewrite the limit: lim_{x→7} ((x - 7)(x + 7))/(x - 7).
    Full step-by-step solution

    Step 1: Direct substitution gives (7² - 49)/(7 - 7) = (49 - 49)/0 = 0/0, an indeterminate form. Step 2: Factor the numerator: x² - 49 = (x - 7)(x + 7). Step 3: Rewrite the limit: lim_{x→7} ((x - 7)(x + 7))/(x - 7). Step 4: Cancel the common factor (x - 7) (valid since x ≠ 7 in the limit): lim_{x→7} (x + 7). Step 5: Evaluate by direct substitution: 7 + 7 = 14. The answer is 14.

  3. lim_(x→∞) (4x³ - 7x + 2)/(2x³ + 5x² - 3) = ? Answer: 2 Solution: Identify the highest power of x in both numerator and denominator. Both are degree 3. Divide every term by x³, the highest power.
    Full step-by-step solution

    Step 1: Identify the highest power of x in both numerator and denominator. Both are degree 3. Step 2: Divide every term by x³, the highest power. Step 3: The expression becomes (4 - 7/x² + 2/x³)/(2 + 5/x - 3/x³). 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.

  4. Aroha is a geologist studying the rate of sediment accumulation in a river delta. The depth of sediment (in meters) is modeled by the function S(t) = (t^3 - 125)/(t - 5) for t ≠ 5, where t represents time in thousands of years since the delta began forming. Due to a core sample gap, the depth at exactly t = 5 thousand years was not recorded. Aroha needs to determine what depth the sediment approaches as time gets arbitrarily close to 5 thousand years from both sides. What is the limit of S(t) as t approaches 5? Answer: 75 Solution: Write the limit we need to find: lim(t→5) (t^3 - 125)/(t - 5) Recognize that 125 = 5^3, so the numerator is t^3 - 5^3, a difference of cubes.
    Full step-by-step solution

    Step 1: Write the limit we need to find: lim(t→5) (t^3 - 125)/(t - 5) Step 2: Recognize that 125 = 5^3, so the numerator is t^3 - 5^3, a difference of cubes. Step 3: Factor the numerator using the difference of cubes formula: a^3 - b^3 = (a - b)(a^2 + ab + b^2). Here, a = t and b = 5, so t^3 - 125 = (t - 5)(t^2 + 5t + 25). Step 4: Rewrite the function: S(t) = [(t - 5)(t^2 + 5t + 25)]/(t - 5) Step 5: Cancel the common factor (t - 5) from numerator and denominator (valid for t ≠ 5): S(t) = t^2 + 5t + 25 Step 6: Evaluate the simplified expression at t = 5: 5^2 + 5(5) + 25 = 25 + 25 + 25 = 75 Step 7: Therefore, the limit as t approaches 5 is 75 The answer is 75.

  5. Sophia is analyzing the graph of the function f(x) = (sqrt(2x + 1) - 1) / (x) for x ≠ 0. On a coordinate plane, the graph appears to have a hole at x = 0. As x approaches 0 from both sides, the y-values approach a specific number. Determine the limit of f(x) as x approaches 0 by using algebraic manipulation. Answer: 1 Solution: Write the limit expression: lim(x→0) (sqrt(2x + 1) - 1) / x.
    Full step-by-step solution

    Step 1: Write the limit expression: lim(x→0) (sqrt(2x + 1) - 1) / x. Step 2: Multiply the numerator and denominator by the conjugate of the numerator, which is sqrt(2x + 1) + 1: = lim(x→0) [(sqrt(2x + 1) - 1)(sqrt(2x + 1) + 1)] / [x(sqrt(2x + 1) + 1)]. Step 3: Simplify the numerator using the difference of squares: (sqrt(2x + 1))^2 - 1^2 = (2x + 1) - 1 = 2x. The expression becomes: lim(x→0) (2x) / [x(sqrt(2x + 1) + 1)]. Step 4: Cancel the common factor x (since x ≠ 0): = lim(x→0) 2 / (sqrt(2x + 1) + 1). Step 5: Now evaluate the limit by direct substitution: 2 / (sqrt(2*0 + 1) + 1) = 2 / (sqrt(1) + 1) = 2 / (1 + 1) = 2 / 2 = 1. The answer is 1.

  6. Hana is an environmental scientist studying the rate of pollutant decay in a lake. The concentration of the pollutant (in parts per million) is modeled by the function C(x) = (4x^2 - 36)/(x - 3) for x ≠ 3, where x represents time in days after a chemical spill. To determine the immediate concentration at the critical time x = 3 days, Hana needs to find what value the concentration approaches as time gets arbitrarily close to 3 days from both sides. What is the limit of C(x) as x approaches 3? Answer: 24 Solution: Write the original function: C(x) = (4x^2 - 36)/(x - 3). Factor the numerator: 4x^2 - 36 = 4(x^2 - 9) = 4(x - 3)(x + 3). Rewrite the function: C(x) = [4(x - 3)(x + 3)]/(x - 3).
    Full step-by-step solution

    Step 1: Write the original function: C(x) = (4x^2 - 36)/(x - 3). Step 2: Factor the numerator: 4x^2 - 36 = 4(x^2 - 9) = 4(x - 3)(x + 3). Step 3: Rewrite the function: C(x) = [4(x - 3)(x + 3)]/(x - 3). Step 4: Cancel the common factor (x - 3) (valid for x ≠ 3): C(x) = 4(x + 3). Step 5: Evaluate the limit as x approaches 3: lim(x→3) 4(x + 3) = 4(3 + 3) = 4(6) = 24. The answer is 24.

  7. Olivia sketches the graph of the function f(x) = (x^2 - 10x + 25) / (x^2 - 25). On the coordinate plane, she notices a hole in the graph at x = 5. As x approaches 5 from both sides, what y-value does the function approach? Answer: 0 Solution: Write the limit: lim(x→5) (x^2 - 10x + 25) / (x^2 - 25). Factor the numerator: x^2 - 10x + 25 = (x - 5)(x - 5) = (x - 5)^2. Factor the denominator: x^2 - 25 = (x - 5)(x + 5).
    Full step-by-step solution

    Step 1: Write the limit: lim(x→5) (x^2 - 10x + 25) / (x^2 - 25). Step 2: Factor the numerator: x^2 - 10x + 25 = (x - 5)(x - 5) = (x - 5)^2. Step 3: Factor the denominator: x^2 - 25 = (x - 5)(x + 5). Step 4: Rewrite the expression: (x - 5)^2 / ((x - 5)(x + 5)). Step 5: Cancel one common factor (x - 5): (x - 5) / (x + 5). Step 6: Substitute x = 5: (5 - 5) / (5 + 5) = 0 / 10 = 0. The answer is 0.