Interpret the behavior of functions using limits involving infinity.
I understand that evaluating a limit at infinity is the same thing as finding the end behavior/horizontal asymptote.
I can use the rules for rational functions to determine a limit at infinity.
I can describe vertical asymptotes and unbounded behavior using infinite limits.
Quick Lesson Plan
This lesson connects students’ prior knowledge about horizontal asymptotes and end behavior with the new concept and notation of finding limits as x approaches infinity. They study a function that models memory over time and compare graphical and analytical representations of this function’s long-run behavior. Emphasis is placed on interpreting situations in context and connecting concepts.
This lesson focuses on rational functions and the end-behavior of two polynomials at x-values near infinity. If presented with a quotient containing other functions (transcendentals), end behavior can often be determined by comparing the dominance of the numerator or denominator using the mnemonic Ten Fabulous Engineers Prefer Learning Calculus. Tower functions (x^x^x) grow faster than Factorials (x!) grow faster than Exponentials (a^x) grow faster than Polynomials (x^n) grow faster than Logarithms grow faster than Constants. Students will learn other methods for evaluating limits for complex functions, but memorizing TFEPLC can guide their reasoning at this stage.
Limits at infinity have appeared on both the multiple choice and free response sections of the exam. Check out this great question! (1998 AB 2a).
Horizontal asymptotes describe the behavior of a function at the ends of the number line. A function may attain this value or cross this horizontal line many times before asymptotic behavior appears.