MTH 251 – Differential Calculus Chapter 2 – Limits and ContinuityWilliam Whewell (1794-1866)What is a Limit?Limit Example: Repeating DecimalsLimits: Notation & MeaningThe First Rule for Calculating Limits !!Estimating Limits by SamplingSlide 8Slide 9Slide 10Calculating Limits by SamplingEstimating Limits from GraphsEstimating Limits – SummaryLimit LawsLimit of a PolynomialLimit of a Rational ExpressionSpecial cases …Sandwich Theorem (aka: Squeeze Theorem or Pinching Theorem)MTH 251 – Differential CalculusChapter 2 – Limits and ContinuitySection 2.2Limit of a Function and Limit LawsCopyright © 2010 by Ron Wallace, all rights reserved.William Whewell (1794-1866)A limit is a peculiar and fundamental concept, the use of which in proving the propositions of Higher Geometry cannot be superseded by any combination of other hypotheses and definition.William Whewell: 19th century Englishman known for his works on the history and philosophy of science. In 1833, at the request of a poet, he invented the term “scientist.”What is a Limit?1. The point, edge, or line beyond which something cannot or may not proceed. 2. A confining or restricting object, agent, or influence. 3. The greatest or least amount, number, or extent allowed or possible.The American Heritage DictionaryLimit Example: Repeating Decimals•Consider the sequence of numbers …0.450.45450.4545450.45454545etc.•Each number in this sequence gets closer and closer to the above number.•That is, the limit of this sequence of numbers is …0.4550.4511=Limits: Notation & Meaning•Read as …“The limit of f(x) as x approaches a is L.”•Intuitive meaning …As values of x are chosen closer and closer to a (but never equal to a), f(x ) becomes closer and closer to (possibly equal to) the number L.Lxfax)(limAlternate Notation: axLxf  as )(The First Rule for Calculating Limits !!•To find …•… calculate•For most functions, as long as f(x) is not a piecewise function and f(a) exists, …lim ( )x af x�( )f alim ( ) ( )x af x f a�=Estimati n g Limits by Sampling•To find …•… calculate f(x) for values of x closer and closer to a and look for a pattern (note: you MUST use values on both sides of a).•Tools? Use a calculator or computer.Warning: Results may be misleading.lim ( )x af x�Estimati n g Limits by SamplingExample 1 of 4•f(1.9) = 11.41•f(1.99) = 11.9401•f(1.999) = 11.994001•f(1.9999) = 11.99940001•f(2.1) = 12.61•f(2.01) = 12.0601•f(2.001) = 12.006001•f(2.0001) = 12.0006000128lim :Determine32xxxCheck values closer and closer to 2 (from above & below).The limit appears to be 12.Estimati n g Limits by SamplingExample 2 of 4Check values closer and closer to 2 (from above & below).The limit appears to be ±.28lim :Determine32xxx•f(1.9) = -148.59•f(1.99) = -1588.06•f(1.999) = -15,988.01•f(1.9999) = -159,988.00•f(2.1) = 172.61•f(2.01) = 1612.06•f(2.001) = 16,012.01•f(2.0001) = 160,012.00Can’t be two values, therefore it does not exist.Estimati n g Limits by SamplingExample 3 of 4Check values closer and closer to 0 (from above & below).The limit appears to be 0.5 or 0 or ???.Look at the graph between 0.005 … it does not exist.xx1coslim :Determine0•f(-/10) = 0.9985•f(-/100) = 0.8496•f(-/1000) = 0.7468•f(-/10000) = 0.5461•f(/10) = 0.9985•f(/100) = 0.8496•f(/1000) = 0.7468•f(/10000) = 0.5461Calculating Limits by SamplingExample 4 of 4Check values closer and closer to 0 (from above & below).The limit appears to be  1.Can’t be two values, therefore it does not exist.xxx 0lim :Determine•f(-0.1) = -1•f(-0.01) = -1•f(-0.001) = -1•f(-0.0001) = -1•f(-0.00001) = -1•f(0.1) = 1•f(0.01) = 1•f(0.001) = 1•f(0.0001) = 1•f(0.00001) = 1Estimati n g Limits from Graphs4lim ( )xf x�-=2lim ( )xf x�-=0lim ( )xf x�=2lim ( )xf x�=3lim ( )xf x�=1lim ( )xf x�=Estimati n g Limits – Summary•Consider the graph of f(x) around x = c•Consider values of the function f(x) near x = c•Substituting - i.e. Evaluate f(c)Each option has its “limit”.(pun somewhat intended)lim ( )x cf x L�=Limit Laws1. Sum Rule2. Difference Rule3. Product Rule4. Constant Multiple Rule5. Quotient Rule6. Power & Root Rulelim ( )Assume: lim ( )x cx cf x Lg x M��==[ ]lim ( ) ( )x cf x g x L M�+ = +[ ]lim ( ) ( )x cf x g x L M�- = -[ ]lim ( ) ( )x cf x g x LM�=[ ]lim ( )x ckf x kL�=( )lim , if M 0( )x cf x Lg x M�� �= �� �� �[ ]lim ( ), & 0If is even, 0rrssx cf x Lr s Z ss L�=ι�These can be proven later after the formal definition of a limit is given (see appendix 4 for details).Limit of a Polynomial•That is, the limit of a polynomial is the value of that polynomial at c.•Example …lim ( ) ( )x cP x P c�=4 22lim5 2 3xx x�- + =11 1 0( ) n nn nP x a x a x a x a--= + + + +gggLimit of a Rational Expression•Note: P(x) and Q(x) are polynomials.•Examples …( ) ( )lim if ( ) 0( ) ( )x cP x P cQ cQ x Q c�= �234lim2xxx�-=-224lim2xxx�-=-Special cases …•Rational expressions where the limit of the denominator is zero.Remove the factor (x – c), if possible.•Rationalizing the numerator or denominator.224lim2xxx�--05 4 2limxxx�+ -2323 10lim8xx xx�+ --Sandwich Theorem(aka: Squeeze Theorem or Pinching Theorem)•If g(x) ≤ f(x) ≤ h(x) for all x near x = c andthenlim ( ) lim ( )x c x cg x h x L� �= =lim ( )x cf x L�=ch(x)g(x)f(x)Example:It can be shown geometrically that …sincos 1xxx� �0sinlim ?xxx�=… therefore