Limit Calculation Worksheets Grade 12

1. lim_{x→0} (sin(3x))/(2x) = ?
2. Tane is a marine biologist studying the population growth of a rare species of fish in a protected lagoon. The population size, in hundreds, is modeled by the function P(t) = (t^3 - 27)/(t - 3) for t ≠ 3, where t represents time in years since the study began. Due to a data collection gap, the population at exactly t = 3 years was not recorded. Tane needs to determine what population value the function approaches as time gets arbitrarily close to 3 years to complete his report. What is the limit of P(t) as t approaches 3?
3. A civil engineer is designing a suspension bridge where the cable's shape follows the function f(x) = (x³ - 8)/(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?

…and 4 more problems

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?
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?
3. Evaluate the limit: lim(x→0) (e^(2x) - 1 - 2x) / (x^2)

…and 3 more problems

1. 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) = (3t^2 - 12)/(t^2 - 4) milligrams per liter, where t is time in hours after administration. As time approaches 2 hours, the function appears to approach an indeterminate form. Determine what concentration the drug approaches as t → 2 hours by simplifying the function algebraically.
2. 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?
3. lim_(x→∞) (5x^3 - 2x^2 + 7)/(3x^3 + 4x - 1) = ?

…and 4 more problems