MTH 251 – Differential Calculus Chapter 2 – Limits and ContinuityLimits Involving InfinitySlide 3Limits as x  Limits as x  –Evaluating Limits as x  ±Limits at Infinity of Rational FunctionsHorizontal AsymptotesOblique AsymptotesWhat is happening when L DNE?Slide 11Slide 12Slide 13Slide 14What happens when x  2 ?Limit Definition Modification …Likewise for negative infinity …A third possibility … consider …Slide 19Limits of Rational FunctionsVertical AsymptotesSlide 22MTH 251 – Differential CalculusChapter 2 – Limits and ContinuitySection 2.6Limits Involving Infinity;Asymptotes of GraphsCopyright © 2010 by Ron Wallace, all rights reserved.Limits Involving Infinity•  ?Not a number – more of a concept or behavior•x  x is increasing without bound and positive•x  – x is decreases without bound and negative•f(x)  ±Does not say the limit exists, it doesn’t.It does say the function increases or decreases without bound as x approaches some value (i.e. describes the behavior)Limits Involving Infinity•Examples, consider …Terminal velocity: time  •Raindrop: ~17 mph•Penny: ~22 mph•Skydiver: ~120 mphThe cost of removal of pollution% of pollution removedCost to remove pollution 100%TimeVelocity (mph)–22… means that for all  > 0 there exist a number M such that if x > M then |f(x) – L| < .Limits as x  lim ( )xf x L��=L + LL - MIntuitively – As x gets bigger, f(x) gets closer and closer to L.… means that for all  > 0 there exist a number N such that if x < N then |f(x) – L| < .Limits as x  –lim ( )xf x L�- �=L + LL - NIntuitively – As x gets smaller, f(x) gets closer and closer to L.•First …•All other limit laws from before are also true for x  ±Evaluating Limits as x  ±1lim 0xx���=Limits at Infinity ofRational Functions•Numerator & Denominator w/ Same DegreeLimit is the ratio of the leading coefficients•Numerator with a Smaller DegreeLimit is zero.•Denominator with a Smaller Degree – later!3 232 5 11lim7 1xx xx x��- ++ -Divide top & bottom by x335 11lim7 1xxx x��++ -Divide top & bottom by x3Horizontal Asymptotes•If … … then y = A is a horizontal asymptote to the right.•If … … then y = B is a horizontal asymptote to the left.•Examplelim ( )xf x A��=lim ( )xf x B�- �=223 5( )2 7x xf xx-=+15Oblique Asymptotes•If …… where the degree of P(x) is one greater then the degree of Q(x), then there will be an oblique (i.e. slant) asymptote.•To find the asymptote, divide Q(x) into P(x) and drop the remainder.•Example( )( )( )P xf xQ x=23 5( )2xf xx-=+lim 0( )xremainderQ x���=What is happening when L DNE?lim ( )x cf x L�=What is happening when L DNE?•f(x) not defined around x = c …lim ( )x cf x L�= 2limxx� -–2What is happening when L DNE?•f(x) not defined around x = c …•Jump …lim ( )x cf x L�= 022lim ( ) 1 , 0 ( )3 , 0xxf xxf xx x�- ��=�- >�What is happening when L DNE?•f(x) not defined around x = c …•Jump …•Oscillation …lim ( )x cf x L�=11 1lim sin xx+� -What is happening when L DNE?•f(x) not defined around x = c …•Jump …•Oscillation …•Increase/Decrease wo/ bound …lim ( )x cf x L�=22 3( 2) 2lim xxx+-�This last case is the topic of the rest of this section.What happens when x  2 ?22 3( 2) 2lim xxx+-�x f(x)2.1 7202.01 70,2002.001 7,002,0002.0001 700,020,0001.9 6801.99 69,8001.999 6,998,0001.9999 699,980,000IncreasesWithout Bound=�NOTE: This is NOT saying the limit exists. It is describing the behavior of the function!Limit Definition Modification …For all B > 0 there is a  > 0 such thatif 0 < |x - c| <  then f(x) > B clim ( ) xf x�IncreasesWithout Bound=�Bc + c – Note that, if extended up, the function will be entirely inside the box.Likewise for negative infinity … clim ( ) xf x�DecreasesWithout Bound=- �–Bc + c – Note that, if extended down, the function will be entirely inside the box.For all B > 0 there is a  > 0 such thatif 0 < |x - c| <  then f(x) < –BA third possibility … consider …•Left hand limit?•Right hand limit?•(Two-Sided) Limit?51 1lim xxx+-�DNE=51 1lim xxx-+-�=- �51 1lim xxx++-�=�Limits at Infinity ofRational Functions•Numerator & Denominator w/ Same DegreeLimit is the ratio of the leading coefficients•Numerator with a Smaller DegreeLimit is zero.•Denominator with a Smaller DegreeLimit is , or –325 11lim7 1xxx x��++ -Revisited!Divide top & bottom by x2Limits of Rational Functions222 816 4lim x xxx- --�222 84 4lim x xxx- --�228 1616 4lim x xxx- +-�24( 4) 4lim xxx+-�27( 4) 4lim xxx--�2716 4lim xxx+-�74 4lim xxx-+�74 4lim xxx+-�00 00a abbVertical Asymptotes•For rational functionsWhere are the vertical asymptotes (if any)?For all values x = c where … q(x) = 0 AND•Example( )( )( )p xf xq x=lim ( )x cf x��=��22 3( )1xf xx-=-Vertical Asymptotes•Other functions w/ vertical asymptotes?Trigonometric FunctionsLogarithm