Outline. Limits as x

Transcription

1 MS: IT Mathematics Limits & Continuity Limits at Infinity John Carroll School of Mathematical Sciences Dublin City University Introduction So far, we have only considered its as c where c is some finite value. We now eamine what happens to functions as becomes infinitely large, i.e. as. Limits (3/4) MS: IT Mathematics John Carroll 3 / 33 Limits (3/4) MS: IT Mathematics John Carroll 4 / 33

2 Simple Illustration One eample to consider first is f () =. -value f ()-value As becomes infinitely large, then becomes smaller and smaller, and we write: Limits (3/4) MS: IT Mathematics John Carroll 5 / 33 Limits (3/4) MS: IT Mathematics John Carroll 6 / 33 A Second (Comparative) Illustration the evaluation of. As becomes large, then becomes small but is even larger than and so is even smaller than. Hence, we conclude that Limits (3/4) MS: IT Mathematics John Carroll 7 / 33 Limits (3/4) MS: IT Mathematics John Carroll 8 / 33

3 More Generally The more general result holds for any n > 0. n Making Comparisons If, instead, we require, we can reason as follows: If is infinitely large, then is also infinitely large, and dividing by an infinitely large number produces an infinitely small number, and so that the it must be zero. We could also proceed as follows: divide above and below by : Note that n can be less than, for eample n = when we may write Now, take the it as : = = 0 Limits (3/4) MS: IT Mathematics John Carroll 9 / 33 Limits (3/4) MS: IT Mathematics John Carroll 0 / 33 Rules for Limits Rules for Limits The General Case Finite Limits as ± The function f has a real it L as tends to if, however small a distance we choose, f () gets closer than this distance to L and stays closer, no matter how large becomes and we write f () = L or f () L, as The function f has a real it L as tends to if, however small a distance we choose, f () gets closer than this distance to L and stays closer, no matter how large and negative becomes and we write f () = L or f () L, as Limits (3/4) MS: IT Mathematics John Carroll / 33 Limits (3/4) MS: IT Mathematics John Carroll / 33

4 Rules for Limits The General Case Rules for Limits The General Case f () = L Rules for Limits as ± If L, M and k are real numbers and f () = L, ± g() = M ± then Limits (3/4) MS: IT Mathematics John Carroll 3 / 33 (i) Sum Rule: (ii) Difference Rule: (iii) Product Rule: f () + g() = L + M. ± ± ± (iv) Constant Multiple Rule: f () g() = L M. f ()g() = LM. kf () = kl. ± f () (v) Quotient Rule: If M 0, then ± g() = L M. (vi) Power Rule: If r and s are integers with no common factors and s 0, then (f ± ())r/s = L r/s, provided that L r/s is a real number. Limits (3/4) MS: IT Mathematics John Carroll 4 / 33 : Illustration : Illustration Illustrations :Limits as ± Simple Rational Function Let f () = and g() = +. Therefore, f () = = g() Set What is h() = f () g() = + h()? Limits (3/4) MS: IT Mathematics John Carroll 5 / 33 Limits (3/4) MS: IT Mathematics John Carroll 6 / 33

5 Plot of + : Illustration Illustrations + : Illustration Illustrations As we are assuming is large (and hence non-zero), we can divide through by (the highest power of occurring in the denominator) to get: h() = + = + As, Therefore 0 and. h() = + = = Limits (3/4) MS: IT Mathematics John Carroll 7 / 33 Limits (3/4) MS: IT Mathematics John Carroll 8 / 33 Limits at Infinity: Worked Eamples Limits at Infinity: Worked Eamples Problem Solving General Approach We will evaluate some its at infinity in the following eamples by dividing above and below by the highest power of in the original epression. Limits (3/4) MS: IT Mathematics John Carroll 9 / 33 Limits (3/4) MS: IT Mathematics John Carroll 0 / 33

6 Limits at Infinity: Worked Eamples Limits at Infinity: Worked Eamples Eample To evaluate 4 we divide above and below by and take its as follows: 4 = 4 0 Eample To evaluate 4 we again divide above and below by and take its as follows: 4 = 4 0 Note Although the numerator becomes infinitely large, the denominator 4 was still infinitely larger than the numerator and so the overall ratio was zero in the it. Limits (3/4) MS: IT Mathematics John Carroll / 33 Limits (3/4) MS: IT Mathematics John Carroll / 33 Limits at Infinity: Worked Eamples Limits at Infinity: Worked Eamples Eample We divide above and below by the highest power which, in this case, is 4 : = Eample We divide above and below by the highest power which, in this case, is 3, to obtain: = = = Limits (3/4) MS: IT Mathematics John Carroll 3 / 33 Limits (3/4) MS: IT Mathematics John Carroll 4 / 33

7 Limits at Infinity: Worked Eamples Limits at Infinity: Worked Eamples Eample 5 To evaluate note that the highest power is 5 and so we obtain: = = 0 = Eample 6 + We divide above and below by the highest power, namely, to obtain: Note that + = + = = + 0 Limits (3/4) MS: IT Mathematics John Carroll 5 / 33 Limits (3/4) MS: IT Mathematics John Carroll 6 / 33 Limits at Infinity: Worked Eamples Limits at Infinity: Worked Eamples Eample We divide above and below by the highest power, 5, to obtain: = = = Eample 8 As +, then certainly > 0 and, when > 0, we have = Hence, the it which we require must be the it of the constant value, i.e. = = Limits (3/4) MS: IT Mathematics John Carroll 7 / 33 Limits (3/4) MS: IT Mathematics John Carroll 8 / 33

8 Limits at Infinity: Worked Eamples = = Limits at Infinity: Worked Eamples 3 Limits (3/4) MS: IT Mathematics John Carroll 9 / 33 Limits (3/4) MS: IT Mathematics John Carroll 30 / 33 Limits at Infinity: A Special Case ( + ) Limits at Infinity: A Special Case Rationalize the Numerator Limits (3/4) MS: IT Mathematics John Carroll 3 / 33 Limits (3/4) MS: IT Mathematics John Carroll 3 / 33

9 Limits at Infinity: A Special Case ( ) Show that + Rationalize the Numerator Rationalize the Numerator ( + ) [ ] ( = + ) ( + ) = + + = + + Limits (3/4) MS: IT Mathematics John Carroll 33 / 33