MATH 234, Calculus for BusinessLecture 5, Textbook Sections 3.1Introduction to LimitsFebruary 1st, 2017AnnouncementsHW2 is due Thursday 2/2, 11:59pm; late HWs are not accepted.Section 3.1, LimitsWhy consider limits?Limits give the prediction of the behavior of a function near a point,even if it is impossible to discuss the behavior at the point itself.Example 1. Let f(x) =x2+ x − 6x2− 3x + 2. Note thatf(2) =which (i.e., x = 2 is of f(x)). Predict behavior near x = 2.Remark. There are two methods to predict behavior near a point/compute limits(1)(2)First focus on Method 1Method 1−→ x approaches 2 from left −→ ←− x approaches 2 from right ←−x 1.9 1.99 1.999 2 2.001 2.01 2.1f(x) undefinedAs x approaches 2 from left, then f (x) .As x approaches 2 from right, then f(x) .As x approaches 2 from both sides, then f (x) .1The graph of f(x) =x2+ x − 6x2− 3x + 2isDefinition (Limits from the left and the right).As x approaches 2 from the left, then f (x) : notation iswhich is read “ ”.As x approaches 2 from the right, then f(x) : notation iswhich is read “ ”.Definition (Two-sided limits, or, simply, limits).As x approaches 2 from both sides, then f (x) : notation iswhich is read “ ”.Remark. For a two-sided limit as x approaches a of f(x) to exist,• Limits as x approaches a from left and right of f(x) must .• The left and right limits must .2Example 2. Compute limx→0−f(x), limx→0+f(x), and limx→0f(x), wheref(x) =|x|x=xxif x > 0undefined if x = 0−xxif x < 0Solution. Note that f (x) = for x > 0 and f(x) = for x < 0 andx -0.1 -0.01 -0.001 0 0.001 0.01 0.1f(x) undefinedSolimx→0−f(x) =andlimx→0−f(x) =The left and right limits , solimx→0f(x)Method 2Theorem 1 (Algebraic Properties of Limits).Fix a and suppose limx→af(x) and limx→ag(x) both exist.(1) If k is constant, then limx→akf(x) = .(2) limx→a(f(x) ± g(x)) = .(3) limx→a(f(x) ∗ g(x)) = .(4) limx→af(x)g(x)= , as long as .(5) If p(x) is a polynomial, then limx→ap(x) = .(6) If k is constant, then limx→af(x)k= .(7) If k is constant and positive, then limx→akf(x)= .(8) If limx→af(x) > 0, then limx→alogb(f(x)) = as long as .3Example 3 (Warm-up 1). Find limx→2f(x), where f (x) = x3+ 2x + 4.Solution.Example 4 (Warm-up 2). Find limx→6f(x), where f (x) = 24x.Solution.Example 5 (Warm-up 3). Find limx→3f(x), where f (x) = log4(253 + x).Solution.Example 6 (Warm-up 4). Find limx→3f(x), where f (x) = log4(−253 + x).Solution.Example 7 (Introductory example). Let f(x) =x2+ x − 6x2− 3x + 2. Compute limx→2f(x) using properties of limits.Solution.(Non-)Existence of LimitsFix the number a. The expression limx→af(x) does not exist if• or• or or .Sometimes, limx→af(x) may look like00∞∞0 ∗ ∞ ∞0000∞1∞∞ − ∞These expressions . Limit exist. More work is needed.These expressions are the indeterminate forms.4Limits at InfinityLimits at infinity predict what happens in the long term.Example 8. A formula for population of pandas in the world isf(t) =8,000 ∗ et/10007 + et/1000where t is the number of years after 2017.What happens to the pandas in the long term? Do they go extinct?Solution.t 0 10 100 1,000 5,000 10,000 15,000 20,000f(t)In the long term, this formula says the pandas .In fact, the population of pandas in the long term.Regarding the population, the graph of f(t) isThe horizontal asymptote is .Definition (Limits at infinity).The population of pandas in the long term: notation iswhich is read “ ”.To understand when t approaches −∞, notation is ,which is read “ ”.5Theorem 2 (Algebraic Properties of Limits at Infinity).Theorem 1 remains true if x → a is replaced with x → ∞ or x → −∞, butthe only exception is , which is false.To replace , let k be a constant, positive real number.(Prop. ) limx→∞x−k= .(Prop. ) If k is , then limx→−∞x−k= .(Prop. ) Both limx→∞xkand limx→−∞xk.(Prop. ) limx→∞e−kx= and limx→−∞ekx= .(Prop. ) Both limx→∞ekxand limx→−∞e−kx.(Prop. ) limx→∞1ln(x)= , but limx→∞ln(x) .To help remember these properties, draw graphs of each function.Example 9 (Warm-up 1). Compute limx→∞f(x), where f (x) =1x2+1x4.SolutionExample 10 (Warm-up 2). Compute limx→∞f(x), where f (x) =1x2+ x4.SolutionExample 11 (Long term behavior of rational functions, part 1).Compute limx→∞f(x), where f (x) =x3+ 7x2+ x4.Solution6The graph of f(x) =x3+ 7x2+ x4isExample 12 (Long term behavior of rational functions, part 2).Compute limx→∞f(x), where f (x) =3x4+ 7x2+ 2x4.SolutionThe graph of f(x) =3x4+ 7x2+ 2x4isExample 13 (Long term behavior of rational functions, part 3).Compute limx→∞f(x), where f (x) =3x5+ 7x2+ 2x4.Solution7The graph of f(x) =3x5+ 7x2+ 2x4isExample 14 (Revisiting pandas). A formula for population of pandas in the world isf(t) =8,000 ∗ et/10007 + et/1000Using algebraic properties of limits,limt→∞8,000 ∗ et/10007 + et/1000=Question 1 (Summarize long term behavior of rational functions).Let p(x) and q(x) be polynomials (and not the zero polynomial).Consider the long-term behavior ofp(x)q(x)in 3 cases:(Case 1) If degree of p(x) < degree of q(x), what is limx→∞p(x)q(x)?(Case 2) If degree of p(x) = degree of q(x), what is limx→∞p(x)q(x)?(Case 3) If degree of p(x) > degree of q(x), what is limx→∞p(x)q(x)?Remark. The answers are given previous computations (Examples 11, 12, and