Math 124. Section 2.6: Limits at infinity & Horizontal Asymptotes. 1 x. lim

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1 Mat 4 Section.6: Limits at infinity & Horizontal Asymptotes Tolstoy, Count Lev Nikolgevic (88-90) A man is like a fraction wose numerator is wat e is and wose denominator is wat e tinks of imself. Te larger te denominator te smaller te fraction. In H. Eves Return to Matematical Circles, Boston: Prindle, Weber and Scmidt, Convince yourself tat 0 You can use te grap, or just realize tat you can get / as close to zero as you want by just picking a large enoug (positive or negative) value for. For eample, you can get / = by taking,,,, Also: 0 : For eample, you can get / = by taking 9,53,809,54,000, More generally: its at infinity of any negative powers of are zero. Even more generally: if f ( ), ten 0 f ( ) (Sortand: 0 ) (On te oter and, for positive powers a, a and a, depending (on wat?) ). A most useful tecnique in computing its at infinity wen te epression inside te it is a fraction involving powers of, is to divide eac term in bot te numerator (top) and denumerator (bottom) of te fraction by an appropriate power of. Tis is because we ave a it of te type (indeterminate), so depending on te particular case, te it migt end up being a finite number (zero or non-zero), +, or. 0 Eample: 0 0 ( )

2 3 3 ( 3 e) 3 e Eample: 5 (5 ) Here, ( 3) [ ( 3)] * 3 e Eample: ( )?? Tis it is of te type - (also indeterminate). Here you need to rationalize first: ( ( ) ( ) )( ) Now we ave a it of te form /, so we can try to use te tecnique above: ( ) / 3. Sometimes you just need to know your functions: Eample: cos( ) does not eist, because cos(ө) keeps cycling troug all values between - and as Ө -. (on te oter and, cos( ) cos( ). Wy? Because does not depend on Ө, so for te purposes of tat it we can treat cos() as a constant and apply te Constant Law) Eample: e 0 Eample: arctan( y) y

3 4. Find all te asymptotes for te function 9 f ( ) 3 a) Horizontal asymptotes are lines y=l suc tat f ( ) L. To find tem, we need to ceck te its of tis function at + and. 9 ( 9) ( 3) 9 Similarly, ceck tat. 3 So tis function as te same orizontal asymptote, y=, at bot. (could ave different asymptotes!) b) Vertical asymptotes are lines =a suc tat f ( ). a NOTE THE DIFFERENCE BETWEEN THE TWO TYPES OF ASYMPTOTES! A rational function suc as tis one can only ave infinite its at te zeroes of te denominator. But we need to ceck carefully, because te its at tese zeros need not be infinite! First, factor te denominator: 3 ( 3)( ). So te values of to ceck are =3 and = -: 3 9 ( 3)( 3) 3 3 ( 3)( ) ( 3) 3 ( ) Hence =3 is not a vertical asymptote (rater, te function as a removable discontinuity at =3) 9 does not eist since: 3 Hence = - is a vertical asymptote Conclusion: tis function is defined for all, ecept =3 and =-. It as one orizontal asymptote: y=, at bot. It as one vertical asymptote: = -. Cool!

4 Section.7: Computing rates of cange (velocities) Recall our motivation for its: to compute instant rates of cange! Rate of Cange See on grap Compute as Eample Average rate of cange of f() between =a and =a+ Instantaneous rate of cange of f() at =a Slope of secant line troug te grap of f() at =a and =a+ Slope of tangent line to te grap of f at =a rise y f ( a ) f ( a) run f ( a ) f ( a) 0 Average speed: If you did 0 miles in 0 minutes, your average speed was mpm (= 60mp) Actual speed: If get a speeding ticket, your speed at te instant you pass te radar gun was faster tan te legal it. Eample : Te following is te grap of te distance of a car from its starting position after t min. a) Wen was its speed zero? b) Was it driving faster at t= or at t=3? c) Wat can you say about its speed pattern? Answers: a) Te speed is zero wen te tangents ave slope zero (are orizontal). Tat is, at about t=3.5 and t=8. b) Te slope of te tangent line is steeper at t= tan at t=3. Tat is, te car is driving faster at t= tan at t=3 minutes. c) Follow te slopes of te tangents along te grap (use a ruler if it elps you visualize it). Te speed starts off large and positive at t=0, decreases to zero at about t=3.5, becomes negative between t=3.5 and t=8 (first decreasing up to about t=5.7, ten increasing towards zero from t=5.7 to t=8), ten becomes positive and increasing again.

5 Eample : Suppose te distance is given by te function d(t)=t +. a) Wat is te slope of te tangent line to te grap of d(t) at t=3? First compute te slope of te secant line: d(3 ) d(3) [(3 ) ] [3 ] 6 Ten take te it as ->0 of te resulting epression in : Te slope of te tangent line is (6 ) b) Wat is te formula for te speed s(t)? First, compute te average velocity over a time interval of minutes starting at t: d( t ) d( t) (( t ) ) ( t ) t t Ten take te it as ->0 and get: s(t)=t.