Limits Concept Worksheets Grade 12

1. 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
2. lim(x→1) (x² - 1)/(x - 1) = ?
3. 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)?

…and 4 more problems

1. Liam is designing a cylindrical water tank for a new building. The tank's volume must be exactly 1000 cubic meters. To minimize material costs, he needs to find the dimensions that minimize the surface area. If the tank has a circular base with radius r meters and height h meters, what ratio h/r minimizes the surface area while maintaining the fixed volume?
2. From the table: x: 1.97, 1.98, 1.99, 2.00, 2.01, 2.02, 2.03 f(x): 7.88, 7.92, 7.96, 8.00, 8.04, 8.08, 8.12 lim(x→2) f(x) = ?
3. Emma is analyzing the temperature change in a chemical reaction. The temperature T(t) in degrees Celsius after t minutes is modeled by the function T(t) = (t^2 - 16)/(t - 4). When examining the temperature behavior as time approaches 4 minutes, she notices the function appears undefined at exactly t=4. What temperature does the reaction approach as time gets closer and closer to 4 minutes?

…and 5 more problems

1. 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) = ?
2. A marine biologist is studying the population growth of a coral reef ecosystem. The population P(t) of a particular coral species after t years is modeled by the function P(t) = (2t² - 8)/(t - 2). When analyzing the population behavior as time approaches 2 years, she notices the function appears undefined at exactly t = 2. What value does the coral population approach as time gets closer and closer to 2 years?
3. 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?

…and 4 more problems