Determine end behavior of polynomial and rational functions

Learning Objectives Define end behavior using formal limit notation. Determine the end behavior of any polynomial function by applying the Leading Term Test. Calculate the limit of a rational function as x approaches positive or negative infinity by dividing by the highest power of x. Use the degree comparison shortcut to quickly find the end behavior of rational functions. Identify the horizontal asymptote of a rational function from its limit at infinity. Distinguish between end behavior that approaches a finite value and end behavior that approaches infinity. Describe the end behavior of a function using both limit notation and graphical descriptions (e.g., 'rises to the right, falls to the left'). Ever wonder what a function's graph does when it flies of...

TermDefinitionExample End BehaviorThe behavior of the graph of a function f(x) as x approaches positive infinity (x → ∞) or negative infinity (x → -∞). It describes where the y-values of the function are headed.For f(x) = x², as x → ∞, f(x) → ∞ (the graph rises on the right). As x → -∞, f(x) → ∞ (the graph rises on the left). Limit at InfinityThe value L that a function f(x) approaches as x increases or decreases without bound. We write this as lim_{x→∞} f(x) = L or lim_{x→-∞} f(x) = L.For f(x) = 1/x, lim_{x→∞} (1/x) = 0. Leading TermIn a polynomial function written in standard form, the term containing the highest power of the variable.In the polynomial P(x) = -4x⁵ + 2x³ - 7, the leading term is -4x⁵. Horizontal AsymptoteA horizontal line y = L that the graph of a function approaches as...

End Behavior of Polynomials (The Leading Term Test) For a polynomial P(x) = a_n x^n + ... + a_0, the end behavior is determined entirely by its leading term: lim_{x→±∞} P(x) = lim_{x→±∞} a_n x^n. To find the end behavior of any polynomial, ignore all terms except the one with the highest power of x. The sign of the leading coefficient (a_n) and whether the degree (n) is even or odd will tell you if the function rises or falls on the far left and far right. Fundamental Limit Rule for Rational Functions For any positive rational number r > 0 and any constant c, lim_{x→±∞} (c / x^r) = 0. This is the most important rule for evaluating limits of rational functions. It states that as x becomes infinitely large (positive or negative), any constant divided by a positive power of...