Computational Aspects of its. Keeping te simple simple. Recall by elementary functions we mean :Polynomials (including linear and quadratic equations) Eponentials Logaritms Trig Functions Rational Functions Inverse Trig Functions Recall from our discussion of continuity in lecture a function is continuous at a point = a if and only if a. f(a) is defined b. c. eists and a = f(a) (Actually c is te definition of continuity as it implies a and b.) a From wat we've learned in Algebra and Trigonometry all of our elementary functions are continuous at every point in tere domains. Terefore is simply f(a) if a is in te a domain. Many tets refer to tis as evaluating its by direct substitution, owever we'll avoid suc terminology. Eg. 6 4 Te main part of tis lecture is concerned wit evaluating its at points wic are not in te domain end evaluating its of piecewise defined functions.. Rational Functions wen 0/0 is obtained. In tis case factor te numerator and denominator, cancel te common factor and try and evaluate te it. Eg. 6 ( ) ( ) ( ) ( ) ( ) ( ) 4 Will tis always work and wy? Recall a rational function is a function of te form f() = p( ) polynomials. q( ) were p() and q() are
recall from Algebra tat if = a is a zero of a polynomial ten (-a) is a factor. Terefore in te case wen we get 0/0 p() and q() ave a common zero and terefore a common factor. (Be careful toug we may still get an infinite it).. Epressions wit radicals wen 0/0 is obtained. In tis case rationalize eiter te numerator or denominator (depending on were te radical is) simplify and try to evaluate te it. Eg. 4 4 4 ( 4) 4 4 ( 4) 4 4 Note on rationalization : Recall from Algebra te conjugate of an epression like a b is a b and vice versa and te conjugate of an epression suc as a b is a b. To rationalize an epression multiply te numerator and denominator by te conjugate of te radical epression (For te uneasily confused : -a can also be factored as a a in wic case we can skip te rationalization stage altogeter) An eample of were suc a it arises is te calculation of te derivative of te square root function. d d 0 = 0 0 0 Computational Aspects of Limits at Infinity and Infinite Limits. Before we get into te details let's consider wat may seem some rater strange matematical devices:. c 0 = + were c cannot be 0.. c = 0 c = 0. To understand wy we make tese definitions we need only understand te function f() = :
0 0-0 0 + 0 0 Limits at + Infinity. Polynomials Te it at + is determined by te igest powered term. Eample 7 4 88 7 7 4 88 7 Eample 6 4 4 8 6 4 4 8 6 6 So wy is tis true? To see wy tis works factor out te igest powered term. For eample consider our first eample 7 4 88 7 ( ) 4 88 6 7 Note te last four terms all decay to zero as goes to infinity since tey are all of te form c/. Te only term not decaying to zero is te leading term.
. Rational Functions Since a rational function is simply te ratio of two polynomials to compute te it at infinity consider te ratio of te igest powered terms of te numerator and denominator. Eample 7 Eample 4 7 4 0 Eample 4 7 4. Radicals As before rationalize. (Note - is not necessarily 0) Eample = 0 For Limits at in we'll be developing a metod called L' Hopital's rule wic will really give us a very easy way of dealing wit its at for functions in general. Infinite Limits Te big difference between its at infinity and infinite its is tat wen we talk about its at infinity we are talking about wat appens to te output as te input increases witout bound wile wen we talk about infinite its we are talking about for wat values of te input does te output increase witout bound. A simple way of keeping tis straigt is tat e its at infinity give us te orizontal asymptotes wereas infinite its give us te vertical asymptotes. Recall f() = / again:
Computational Aspects Of Infinite Limits IF a doesn't eist. c 0 ten consider one-side its to see if te it is infinity, negative infinity or Tis last statement needs a bit of eplanation. If te it is infinite one migt argue te it does not eist as infinity is not a number. Consider te functions below: Note for /(-) f() approaces infinity from one-side and negative infinity from te oter. Since te one-sided its are different We would say te it does not eist. For /(-) f() approaces positive infinity from bot sides terefore since te one-sided its are equal we would say te it is infinity. Yes we do want to distinguis between tese cases. Tree Very Important Limits Te derivatives of all te trig functions and eponentials come from basic its
.. sin( ) 0 cos ( ) 0. e 0 0 Once we ave te Proof of te first te second is automatic so Let's start by proving Suppose we consider a segment of te unit circle ten te area of te inscribed triangle is less tan te area of te circular sector subtended by te angle. ( ) sin( ) ( ) from wic it follows sin( ). Here te area of te circular sector is less tan te area of te triangle
( ) but tan() = so ( ) sin( ) cos ( ) from wic it follows cos ( ) sin( ). Combining te results it follows cos ( ) sin( ) cos ( ) 0 Terefore by te squeezing teorem 0 sin( ). 0 To prove # it is just an eercise in Trig Identities : cos ( ) ( cos ( ) ) ( cos ( ) 0 0 ( cos ( ) ) cos ( ) 0 ( cos ( ) ) sin( ) sin( ) ( 0 0) 0 ( cos ( ) Using te results from and we can prove te derivative formula for sin(). dsin( ) sin( ) sin( ) sin( ) cos ( ) sin( ) cos ( ) sin( ) d 0 0 = sin( ) ( ( cos ( ) ) ) 0 sin( ) cos ( ) ( 0 cos ( ) ) cos ( ) 0 As an Eercise Prove dcos ( ) d sin( ) To prove formally we will ave to wait for our differentiation teorem on Inverses However we can consider te following Grap and by te Squeezing Tm # follows. as e is squeezed between - and + bot wic converge to as goes to 0. e 0. 0 0.
. Piecewise Functions Recall a piecewise function is a function of te form: f ( ) a 0 a f() = f ( ) a a.. f n ( ) a n a n Were we can ave any number of intervals. For lack of better terminology let's refer to te a i were te function canges form as break points. To evaluate its at a break point consider te one sided its at te break point. 0 f() = cos ( ) 0 Suppose we want to calculate 0-0 -. 0 cos ( ) 0 + 0 + Terefore = 0 Suppose we want to calculate.
cos ( ) 0 - - ( ) + + Terefore Does not eist. Also it is useful to understand its at infinity and infinite its of some of our elementary functions. Power Functions n n n if n is even if n is odd
. Eponentials 0 e e 0 e e. Te Natural Logaritm 0 + ln( ) ln( ) ln( ) 0 4
4. Tan - () arctan ( ) arctan ( ) atan( ). 0. 0 0. 0 0. Important Point Not to Miss Te its at infinity and negative are te HORIZONTAL ASYMPTOTES OF THE FUNCTION. Note te Arctangent Function as a different Asymptote at negative infinity tan it as at infinity. We call tese alf-assymptotes. (ok not really)