A Silly Game A particularly silly game which two people can play is “who can say the highest number”. This is a back-and-forth game just like the “choose delta” game which we just saw. Every time your oppenent says a number, you can just say the number that is one higher than their number to stay in the game! Adrian Dudek The Formal Definition of a Limit 13 / 37
A Silly Game A particularly silly game which two people can play is “who can say the highest number”. This is a back-and-forth game just like the “choose delta” game which we just saw. Every time your oppenent says a number, you can just say the number that is one higher than their number to stay in the game! However, rather than playing in this silly game, you could simply program a robot to respond for you with x + 1, where x is the number your opponent chooses. Adrian Dudek The Formal Definition of a Limit 13 / 37
A Silly Game A particularly silly game which two people can play is “who can say the highest number”. This is a back-and-forth game just like the “choose delta” game which we just saw. Every time your oppenent says a number, you can just say the number that is one higher than their number to stay in the game! However, rather than playing in this silly game, you could simply program a robot to respond for you with x + 1, where x is the number your opponent chooses. We want to do the same thing with epsilons and deltas! Adrian Dudek The Formal Definition of a Limit 13 / 37
Limits: Example Let’s get back to our example! So we wish to keep 4 − ǫ < x + 4 < 4 + ǫ by keeping − δ < x < δ . Adrian Dudek The Formal Definition of a Limit 14 / 37
Limits: Example Let’s get back to our example! So we wish to keep 4 − ǫ < x + 4 < 4 + ǫ by keeping − δ < x < δ . We simply do the same algebra we did to the specific examples: Add − 4 to all sides of 4 − ǫ < x + 4 < 4 + ǫ to get: − ǫ < x < ǫ Adrian Dudek The Formal Definition of a Limit 14 / 37
Limits: Example Let’s get back to our example! So we wish to keep 4 − ǫ < x + 4 < 4 + ǫ by keeping − δ < x < δ . We simply do the same algebra we did to the specific examples: Add − 4 to all sides of 4 − ǫ < x + 4 < 4 + ǫ to get: − ǫ < x < ǫ So in this case, ǫ = δ . Adrian Dudek The Formal Definition of a Limit 14 / 37
Limits: Example Let’s get back to our example! So we wish to keep 4 − ǫ < x + 4 < 4 + ǫ by keeping − δ < x < δ . We simply do the same algebra we did to the specific examples: Add − 4 to all sides of 4 − ǫ < x + 4 < 4 + ǫ to get: − ǫ < x < ǫ So in this case, ǫ = δ . This is all we have to do to prove a limit! Provide a response δ in terms of any ǫ . Of course, there is a little bit more to write out, but the hard work is done! Adrian Dudek The Formal Definition of a Limit 14 / 37
Limits: Graphical Example Suppose we have a function f ( x ), and we wish to show that lim x → 3 f ( x ) = 5. Adrian Dudek The Formal Definition of a Limit 15 / 37
Limits: Graphical Example Here ǫ = 1, and so we must choose a δ which works. We can see that δ = 1 is a fine choice. Adrian Dudek The Formal Definition of a Limit 16 / 37
Limits: Graphical Example Here ǫ = 1 / 2, and so we must choose a δ which works. We can see that δ = 1 / 2 is a fine choice. Adrian Dudek The Formal Definition of a Limit 17 / 37
Limits: Graphical Example Here ǫ = 0 . 25, and so we must choose a δ which works. We can see that δ = 0 . 25 is a fine choice. Adrian Dudek The Formal Definition of a Limit 18 / 37
Limits: Example Example: Prove x → 2 3 x + 4 = 10 lim That is, show that as x gets really close to 2, then 3 x + 4 gets really close to 10. Adrian Dudek The Formal Definition of a Limit 19 / 37
Limits: Example We want 3 x + 4 to be really close to 10. We do this by specifying that the distance between them remain less than any positive number ǫ : | 3 x + 4 − 10 | < ǫ Adrian Dudek The Formal Definition of a Limit 20 / 37
Limits: Example We want 3 x + 4 to be really close to 10. We do this by specifying that the distance between them remain less than any positive number ǫ : | 3 x + 4 − 10 | < ǫ We want to show that this can be accomplished by keeping the distance between x and 2 less than any amount δ : | x − 2 | < δ Adrian Dudek The Formal Definition of a Limit 20 / 37
Limits: Example We want 3 x + 4 to be really close to 10. We do this by specifying that the distance between them remain less than any positive number ǫ : | 3 x + 4 − 10 | < ǫ We want to show that this can be accomplished by keeping the distance between x and 2 less than any amount δ : | x − 2 | < δ The problem is solved by establishing an answer δ in terms of ǫ , so that you have an answer for any ǫ they throw at you! Adrian Dudek The Formal Definition of a Limit 20 / 37
Limits: Example We usually proceed by rearranging the demand | 3 x + 4 − 10 | < ǫ Adrian Dudek The Formal Definition of a Limit 21 / 37
Limits: Example We usually proceed by rearranging the demand | 3 x + 4 − 10 | < ǫ into an inequality of the form | x − 2 | < δ . Then we can simply read off what δ must be. Adrian Dudek The Formal Definition of a Limit 21 / 37
Limits: Example We usually proceed by rearranging the demand | 3 x + 4 − 10 | < ǫ into an inequality of the form | x − 2 | < δ . Then we can simply read off what δ must be. We start by writing the above without absolute value brackets. − ǫ < 3 x + 4 − 10 < ǫ Adrian Dudek The Formal Definition of a Limit 21 / 37
Limits: Example We usually proceed by rearranging the demand | 3 x + 4 − 10 | < ǫ into an inequality of the form | x − 2 | < δ . Then we can simply read off what δ must be. We start by writing the above without absolute value brackets. − ǫ < 3 x + 4 − 10 < ǫ Simplifying slightly we get − ǫ < 3 x − 6 < ǫ Adrian Dudek The Formal Definition of a Limit 21 / 37
Limits: Example − ǫ < 3 x − 6 < ǫ Dividing by 3 we get − ǫ/ 3 < x − 2 < ǫ/ 3 Adrian Dudek The Formal Definition of a Limit 22 / 37
Limits: Example − ǫ < 3 x − 6 < ǫ Dividing by 3 we get − ǫ/ 3 < x − 2 < ǫ/ 3 which is the same as | x − 2 | < ǫ/ 3 Adrian Dudek The Formal Definition of a Limit 22 / 37
Limits: Example − ǫ < 3 x − 6 < ǫ Dividing by 3 we get − ǫ/ 3 < x − 2 < ǫ/ 3 which is the same as | x − 2 | < ǫ/ 3 Thus, for any ǫ > 0 we choose, we would set δ = ǫ/ 3. Adrian Dudek The Formal Definition of a Limit 22 / 37
Limits: Example − ǫ < 3 x − 6 < ǫ Dividing by 3 we get − ǫ/ 3 < x − 2 < ǫ/ 3 which is the same as | x − 2 | < ǫ/ 3 Thus, for any ǫ > 0 we choose, we would set δ = ǫ/ 3. That is, by keeping | x − 2 | < δ = ǫ/ 3, we guarantee that | 3 x + 4 − 10 | < ǫ . Adrian Dudek The Formal Definition of a Limit 22 / 37
Limits: Example That is, by keeping | x − 2 | < δ = ǫ/ 3, we guarantee that | 3 x + 4 − 10 | < ǫ . Adrian Dudek The Formal Definition of a Limit 23 / 37
Limits: Example That is, by keeping | x − 2 | < δ = ǫ/ 3, we guarantee that | 3 x + 4 − 10 | < ǫ . We check as follows: | 3 x + 4 − 10 | = | 3 x − 6 | Adrian Dudek The Formal Definition of a Limit 23 / 37
Limits: Example That is, by keeping | x − 2 | < δ = ǫ/ 3, we guarantee that | 3 x + 4 − 10 | < ǫ . We check as follows: | 3 x + 4 − 10 | = | 3 x − 6 | = 3 | x − 2 | Adrian Dudek The Formal Definition of a Limit 23 / 37
Limits: Example That is, by keeping | x − 2 | < δ = ǫ/ 3, we guarantee that | 3 x + 4 − 10 | < ǫ . We check as follows: | 3 x + 4 − 10 | = | 3 x − 6 | = 3 | x − 2 | Adrian Dudek The Formal Definition of a Limit 23 / 37
Limits: Example That is, by keeping | x − 2 | < δ = ǫ/ 3, we guarantee that | 3 x + 4 − 10 | < ǫ . We check as follows: | 3 x + 4 − 10 | = | 3 x − 6 | = 3 | x − 2 | < 3 ǫ/ 3 Adrian Dudek The Formal Definition of a Limit 23 / 37
Limits: Example That is, by keeping | x − 2 | < δ = ǫ/ 3, we guarantee that | 3 x + 4 − 10 | < ǫ . We check as follows: | 3 x + 4 − 10 | = | 3 x − 6 | = 3 | x − 2 | < 3 ǫ/ 3 < ǫ Adrian Dudek The Formal Definition of a Limit 23 / 37
Limits: The Definition In general, if we wish to show that x → a f ( x ) = L lim Adrian Dudek The Formal Definition of a Limit 24 / 37
Limits: The Definition In general, if we wish to show that x → a f ( x ) = L lim then we need to show that the distance between f ( x ) and L can be made as small as we want, by making the distance between x and a sufficiently small. Adrian Dudek The Formal Definition of a Limit 24 / 37
Limits: The Definition In general, if we wish to show that x → a f ( x ) = L lim then we need to show that the distance between f ( x ) and L can be made as small as we want, by making the distance between x and a sufficiently small. That is, if somebody wants the distance between f ( x ) and L to be less than ǫ , then we need to show that there is some δ (in terms of ǫ ) where keeping the distance between x and a less than δ guarantees this. Adrian Dudek The Formal Definition of a Limit 24 / 37
Limits: The Definition In general, if we wish to show that x → a f ( x ) = L lim then we need to show that the distance between f ( x ) and L can be made as small as we want, by making the distance between x and a sufficiently small. That is, if somebody wants the distance between f ( x ) and L to be less than ǫ , then we need to show that there is some δ (in terms of ǫ ) where keeping the distance between x and a less than δ guarantees this. ∀ ǫ > 0 ∃ δ > 0 s.t | x − a | < δ ⇒ | f ( x ) − L | < ǫ Adrian Dudek The Formal Definition of a Limit 24 / 37
Limits: The Definition Definition: We say that the limit of f ( x ) as x → a is L if ∀ ǫ > 0 ∃ δ > 0 s.t | x − a | < δ ⇒ | f ( x ) − L | < ǫ Adrian Dudek The Formal Definition of a Limit 25 / 37
Limits: The Definition Definition: We say that the limit of f ( x ) as x → a is L if ∀ ǫ > 0 ∃ δ > 0 s.t | x − a | < δ ⇒ | f ( x ) − L | < ǫ You need to remember this for tests and exams. Feel free to recite it at parties to test your memory! Adrian Dudek The Formal Definition of a Limit 25 / 37
Infinite Limits Sometimes we deal with limits as x → ±∞ . One such example is: � 2 + 4 � lim = 2 x x →∞ This says, that as x gets really large, 2 + 4 x gets really close to 2. Adrian Dudek The Formal Definition of a Limit 26 / 37
Infinite Limits Sometimes we deal with limits as x → ±∞ . One such example is: � 2 + 4 � lim = 2 x x →∞ This says, that as x gets really large, 2 + 4 x gets really close to 2. Once again, the way we prove this is the same! You want to show that the distance between 2 + 4 x and 2 can be made smaller than any positive number ǫ , by making x larger than a corresponding number N . Adrian Dudek The Formal Definition of a Limit 26 / 37
Infinite Limits: Example Example: Prove that � 2 + 4 � lim = 2 x x →∞ Adrian Dudek The Formal Definition of a Limit 27 / 37
Infinite Limits: Example Example: Prove that � 2 + 4 � lim = 2 x x →∞ � 2 + 4 � � � � We want to keep x − 2 � < ǫ . How large does x need to be to guarantee this? That � � is, find a number N , where keeping x > N will guarantee this. Adrian Dudek The Formal Definition of a Limit 27 / 37
Infinite Limits: Example Example: Prove that � 2 + 4 � lim = 2 x x →∞ � � 2 + 4 � � � We want to keep x − 2 � < ǫ . How large does x need to be to guarantee this? That � � is, find a number N , where keeping x > N will guarantee this. Once again, we rearrange our original inequality for x . Adrian Dudek The Formal Definition of a Limit 27 / 37
Infinite Limits: Example We start with � � � 2 + 4 � � x − 2 � < ǫ � � Adrian Dudek The Formal Definition of a Limit 28 / 37
Infinite Limits: Example We start with � � � 2 + 4 � � x − 2 � < ǫ � � which gives − ǫ < 4 x < ǫ Adrian Dudek The Formal Definition of a Limit 28 / 37
Infinite Limits: Example We start with � � � 2 + 4 � � x − 2 � < ǫ � � which gives − ǫ < 4 x < ǫ Multiplying through by x (keeping the inequality signs as they are, because x is positive) gives − ǫ x < 4 < ǫ x Adrian Dudek The Formal Definition of a Limit 28 / 37
Infinite Limits: Example We start with � � � 2 + 4 � � x − 2 � < ǫ � � which gives − ǫ < 4 x < ǫ Multiplying through by x (keeping the inequality signs as they are, because x is positive) gives − ǫ x < 4 < ǫ x The right hand side of this says that 4 < ǫ x . We rearrange this to get x > 4 ǫ . Adrian Dudek The Formal Definition of a Limit 28 / 37
Infinite Limits: Example We start with � � � 2 + 4 � � x − 2 � < ǫ � � which gives − ǫ < 4 x < ǫ Multiplying through by x (keeping the inequality signs as they are, because x is positive) gives − ǫ x < 4 < ǫ x The right hand side of this says that 4 < ǫ x . We rearrange this to get x > 4 ǫ . So N = 4 ǫ . Adrian Dudek The Formal Definition of a Limit 28 / 37
Infinite Limits: Example � � � 2 + 4 � < ǫ whenever x > N = 4 � � So, we claim that x − 2 ǫ . � � Adrian Dudek The Formal Definition of a Limit 29 / 37
Infinite Limits: Example � � � 2 + 4 � < ǫ whenever x > N = 4 � � So, we claim that x − 2 ǫ . � � To finish off the proof nicely we will show the first statement is true under the assumption of the second : Adrian Dudek The Formal Definition of a Limit 29 / 37
Infinite Limits: Example � � � 2 + 4 � < ǫ whenever x > N = 4 � � So, we claim that x − 2 ǫ . � � To finish off the proof nicely we will show the first statement is true under the assumption of the second : � � � � � 2 + 4 4 � � � � x − 2 � = � � � � x � � Adrian Dudek The Formal Definition of a Limit 29 / 37
Infinite Limits: Example � � � 2 + 4 � < ǫ whenever x > N = 4 � � So, we claim that x − 2 ǫ . � � To finish off the proof nicely we will show the first statement is true under the assumption of the second : � � � � � 2 + 4 4 � � � � x − 2 � = � � � � x � � The second statement re-arranges to give 4 x < ǫ. Adrian Dudek The Formal Definition of a Limit 29 / 37
Infinite Limits: Example � � � 2 + 4 � < ǫ whenever x > N = 4 � � So, we claim that x − 2 ǫ . � � To finish off the proof nicely we will show the first statement is true under the assumption of the second : � � � � � 2 + 4 4 � � � � x − 2 � = � � � � x � � The second statement re-arranges to give 4 x < ǫ. Hence � � � � � 2 + 4 4 � � � � x − 2 � = � � � � x � � Adrian Dudek The Formal Definition of a Limit 29 / 37
Infinite Limits: Example � � � 2 + 4 � < ǫ whenever x > N = 4 � � So, we claim that x − 2 ǫ . � � To finish off the proof nicely we will show the first statement is true under the assumption of the second : � � � � � 2 + 4 4 � � � � x − 2 � = � � � � x � � The second statement re-arranges to give 4 x < ǫ. Hence � � � � � 2 + 4 4 � � � � x − 2 � = � < | ǫ | = ǫ (since ǫ > 0). � � � � x � Adrian Dudek The Formal Definition of a Limit 29 / 37
Infinite Limits: Example � � � 2 + 4 � < ǫ whenever x > N = 4 � � So, we claim that x − 2 ǫ . � � To finish off the proof nicely we will show the first statement is true under the assumption of the second : � � � � � 2 + 4 4 � � � � x − 2 � = � � � � x � � The second statement re-arranges to give 4 x < ǫ. Hence � � � � � 2 + 4 4 � � � � x − 2 � = � < | ǫ | = ǫ (since ǫ > 0). � � � � x � We have proven the limit. Adrian Dudek The Formal Definition of a Limit 29 / 37
Infinite Limits: The Definition Definition: We say that the limit of f ( x ) as x → ∞ is L if ∀ ǫ > 0 ∃ N > 0 s.t x > N ⇒ | f ( x ) − L | < ǫ Adrian Dudek The Formal Definition of a Limit 30 / 37
Infinite Limits: The Definition Definition: We say that the limit of f ( x ) as x → ∞ is L if ∀ ǫ > 0 ∃ N > 0 s.t x > N ⇒ | f ( x ) − L | < ǫ Exercise: Try to write out the definition of a limit as x → −∞ . Adrian Dudek The Formal Definition of a Limit 30 / 37
Limit of a Sequence What about limits of sequences? We might want to show that the sequence 1 , 1 / 2 , 1 / 3 , 1 / 4 . . . converges to 0. Adrian Dudek The Formal Definition of a Limit 31 / 37
Limit of a Sequence The limit of a sequence requires the same sort of approach as for infinite limits. Adrian Dudek The Formal Definition of a Limit 32 / 37
Limit of a Sequence The limit of a sequence requires the same sort of approach as for infinite limits. If we believe a sequence a n → L , then we must show that we can keep a n arbitrarily close to L , by starting our sequence far enough to the right. Adrian Dudek The Formal Definition of a Limit 32 / 37
Limit of a Sequence The limit of a sequence requires the same sort of approach as for infinite limits. If we believe a sequence a n → L , then we must show that we can keep a n arbitrarily close to L , by starting our sequence far enough to the right. Starting our sequence far enough to the right means that the index n of our sequence a n commences after some positive number N . Adrian Dudek The Formal Definition of a Limit 32 / 37
Limit of a Sequence: Example Let ( a n ) n ≥ 1 be the sequence defined by a n = 1 / (1 + n ). Show that n →∞ a n = 0 lim Adrian Dudek The Formal Definition of a Limit 33 / 37
Limit of a Sequence: Example Let ( a n ) n ≥ 1 be the sequence defined by a n = 1 / (1 + n ). Show that n →∞ a n = 0 lim We want to show that | 1 / (1 + n ) − 0 | < ǫ whenever we keep n > N for some positive number N . Adrian Dudek The Formal Definition of a Limit 33 / 37
Limit of a Sequence: Example Let ( a n ) n ≥ 1 be the sequence defined by a n = 1 / (1 + n ). Show that n →∞ a n = 0 lim We want to show that | 1 / (1 + n ) − 0 | < ǫ whenever we keep n > N for some positive number N . This will work just as before! We want to find N in terms of ǫ . Adrian Dudek The Formal Definition of a Limit 33 / 37
Limit of a Sequence: Example We start with | 1 / (1 + n ) − 0 | < ǫ Adrian Dudek The Formal Definition of a Limit 34 / 37
Limit of a Sequence: Example We start with | 1 / (1 + n ) − 0 | < ǫ which gives − ǫ < 1 / (1 + n ) < ǫ. Adrian Dudek The Formal Definition of a Limit 34 / 37
Limit of a Sequence: Example We start with | 1 / (1 + n ) − 0 | < ǫ which gives − ǫ < 1 / (1 + n ) < ǫ. Multiplying through by (1 + n ) gives: − ǫ (1 + n ) < 1 < ǫ (1 + n ) Adrian Dudek The Formal Definition of a Limit 34 / 37
Limit of a Sequence: Example We start with | 1 / (1 + n ) − 0 | < ǫ which gives − ǫ < 1 / (1 + n ) < ǫ. Multiplying through by (1 + n ) gives: − ǫ (1 + n ) < 1 < ǫ (1 + n ) Dividing through by ǫ gives: − (1 + n ) < 1 /ǫ < 1 + n Adrian Dudek The Formal Definition of a Limit 34 / 37
Limit of a Sequence: Example We start with | 1 / (1 + n ) − 0 | < ǫ which gives − ǫ < 1 / (1 + n ) < ǫ. Multiplying through by (1 + n ) gives: − ǫ (1 + n ) < 1 < ǫ (1 + n ) Dividing through by ǫ gives: − (1 + n ) < 1 /ǫ < 1 + n Adrian Dudek The Formal Definition of a Limit 34 / 37
Limit of a Sequence: Example − (1 + n ) < 1 /ǫ < 1 + n Now rearranging the rightmost part of this inequality gives us n > 1 /ǫ − 1. Adrian Dudek The Formal Definition of a Limit 35 / 37
Limit of a Sequence: Example − (1 + n ) < 1 /ǫ < 1 + n Now rearranging the rightmost part of this inequality gives us n > 1 /ǫ − 1. So, to keep | 1 / (1 + n ) − 0 | < ǫ , we need to start the sequence off after N = 1 /ǫ − 1. We show that this works as follows: Adrian Dudek The Formal Definition of a Limit 35 / 37
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