2 LIMITS

2.2 THE LIMIT OF A FUNCTIONSUGGESTED TIME AND EMPHASIS1 class Essential materialPOINTS TO STRESS1. The various meanings of “limit” (descriptive, numeric, graphic), both finite and infinite. Note thatalgebraic manipulations are not yet emphasized.2. The geometric and limit definitions of vertical asymptotes.3. The advantages and disadvantages of using a graphing device to compute a limit.QUIZ QUESTIONS• Text Question: What is the difference between the statements “ f (a) = L” and “ lim f (x) = L”?x→aAnswer: Thefirstisastatementaboutthevalueof f at the point x = a, the second is a statement aboutthe values of f at points near, but not equal to, x = a.• Drill Question: The graph of a function f is shown below. Are the following statements about f true orfalse? Why?(a) x = a is in the domain of f (b) lim f (x) exists (c) lim f (x) is equal to lim f (x)x→a x→a + x→a −y0 axAnswer:(a) True, because f is defined at x = a.(b) True, because as x gets close to a, f (x) approaches a value.(c) True, because the same value is approached from both directions.MATERIALS FOR LECTURE• Present the “motivational definition of limit”: We say that lim f (x) = L if as x gets close to a, f (x) getsx→aclose to L, and then lead into the definition in the text. (Note that limits will be defined more precisely inSection 2.4.)• Describe asymptotes verbally, and then give graphic and limit definitions. If foreshadowing horizontalasymptotes, note that a function can cross a horizontal asymptote. Perhaps foreshadow the notion of slantasymptotes, which are covered later in the text.• Discuss how we can rephrase the last section’s concept as “the slope of the tangent line is the limit of theslopes of the secant lines as x → 0”.42