Limits at Infinity (Lesson 8.5)
Unit 8 - Day 14
Describe vertical and horizontal asymptotes using limit notation
Evaluate limits as x approaches infinity by comparing growth rates of numerator and denominator functions
Quick Lesson Plan
In today’s lesson, students look at the population growth of an insect species in the Amazon rainforest. Particularly, they are interested in what would happen to the population “after many, many years,” which we always use as code for end behavior. Students use a table and graph to reason that the population is nearing 2500, that y=2500 is the horizontal asymptote, and that the population will not exceed 2500. Students in biology enjoyed connecting this to the concept of carrying capacity. Ask students why it makes sense that the population does not just keep growing indefinitely. The idea of constraints and limiting factors is an important one to note, since a horizontal asymptote denotes that a limit at infinity exists .
Instead of just telling students to divide the leading coefficients, we wanted students to reason informally about rates of change. Even though both the numerator and denominator function are linear, the numerator grows at a rate 2500 times that of the denominator.
As you are monitoring students, ask groups what the y-intercept of P(t) is. This is not one of the learning targets of today, but it’s important to continue to reinforce how graph behavior is reflected in the equation (in this case the ratio of the constant terms, since t=0) and context. Students should recognize this also coincides with the information about 30 insects being transferred to the protected area.
We remind students that they already know a great deal about horizontal asymptotes (see Lesson 2.6 Day 1). It is worthwhile, however, to discuss again how rational functions behave. Particularly, I ask students to reason about the values of fractions. When the numerator of a fraction is large, what happens to the value of the fraction? (It gets large). When the denominator of a fraction is large, what happens to the value of the fraction? (It gets very small). Note that because t is in both the numerator and denominator, both are going to infinity! This should provide some cognitive dissonance for your students! Is the value of the function getting bigger or smaller?? The conclusion is that we must see which function is going to infinity faster. Note that infinity/infinity is another indeterminate form that sends “mixed messages” about the behavior of the function. Remind students that 0/0 was equally ambiguous.
The Important Ideas can be co-constructed as you ask students to think through various scenarios of functions in the numerator and denominator. Though we focus on polynomials, students should recognize that exponential functions grow faster than polynomials of any degree.
If you also teach Calculus, you may have noticed the nod to L’hospital’s rule in the discussion of comparing growth rates.