BUILDING UP VIRTUAL MATHEMATICS LABORATORY Partnership project LLP-2009-LEO- М P-09, MP 09-05414 Limits at Infinity, Asymptotes and Dominant terms 1/38 ------------------ Snezhana Gocheva-Ilieva, Plovdiv University ---------------------
Overview 1. Limits as x →±∞ 2. Basic example: limits at infinity of f ( x )=1/ x 3. Limits laws as x →±∞ 4. Examples using limits laws at ±∞ 5. Remarkable limits at ±∞ 6. Infinite limits at x → a 7. Examples on infinite limits at x → a 8. Asymptotes of the graph 9. Horizontal asymptote 10. Vertical asymptote 11. Oblique asymptote 12. Computer explorations 13. Dominant terms References 2/38 ------------------ Snezhana Gocheva-Ilieva, Plovdiv University ---------------------
1. Limits as x →±∞ In mathematics, the symbol for infinity is indicated as ∞ . It is not a real number. When use ∞ or + ∞ , this means that the considered values become increasingly large positive numbers. When use −∞ , this means that the values become decreasingly large negative numbers. In this lesson we will consider functions defined on unbounded intervals like ( −∞ , a], [a, ∞ ) or ( −∞ , ∞ ). 3/38 ------------------ Snezhana Gocheva-Ilieva, Plovdiv University ---------------------
By analogy with functions on finite intervals it is possible that the function values are bounded when the argument x approaches infinity (written as x →∞ , or x → −∞ , or x →±∞ ). In many cases the function values can approach a finite number, called limit. Definition 1. A function f ( x ) has the limit A as x approaches infinity, noted by = lim ( ) f x A →∞ x if, for every number ε > 0, there exists a corresponding number M such that for all x > M follows | f ( x ) − A | < ε . 4/38 ------------------ Snezhana Gocheva-Ilieva, Plovdiv University ---------------------
Definition 2. A function f ( x ) has the limit A as x approaches minus infinity, noted by = lim ( ) f x A →−∞ x if, for every number ε > 0, there exists a corresponding number M such that for all x < M follows | f ( x ) − A | < ε . 5/38 ------------------ Snezhana Gocheva-Ilieva, Plovdiv University ---------------------
Definition 3. If for a function f ( x ) no limit exists as x approaches + ∞ or −∞ , but all corresponding values increase (decrease) infinitely to + ∞ (or −∞ ) we will say formally that the function limit is + ∞ (or −∞ ), call it and denote as infinite limit = ∞ , lim = −∞ , lim ( ) ( ) f x f x →∞ →∞ x x = ∞ , lim = −∞ . lim ( ) ( ) f x f x →−∞ →−∞ x x 6/38 ------------------ Snezhana Gocheva-Ilieva, Plovdiv University ---------------------
1 ( ) = f x 2. Basic example: limits at infinity of x x ≠ 0 This function is defined for all . We have: 1 1 = , = lim 0 lim 0 →∞ →−∞ x x x x According to the Definition 1, we fix some ε > Proof. 0 and we seek for a corresponding M such that for A = 0 and all x > M we will have 1 1 1 − = − = < ε > ( ) 0 f x A x , from where . ε x x 1 > As x → ∞ it is enough to take any M . ε The second limit can be proved analogically for 7/38 ------------------ Snezhana Gocheva-Ilieva, Plovdiv University ---------------------
1 < − . x → −∞ by taking M ε The behavior of the = lim 0.0001 function is given in Fig.1. f (x )=1/x 0.00005 It shows that: lim = 0 lim = 0 0 f ( x ) decreases to 0 100000 50000 50000 100000 when x →∞ with 0.00005 positive values 0.0001 = - lim f ( x ) increases to 0 Fig. 1 Graphics of f ( x ) =1/ x . when x →−∞ with negative values. 8/38 ------------------ Snezhana Gocheva-Ilieva, Plovdiv University ---------------------
3. Limits laws as x →±∞ Let A , B and λ are real numbers and there exist the = = limits: lim ( ) and lim ( ) f x A g x B . Then →±∞ →±∞ x x { } λ = λ lim ( ) f x A 1. Constant multiple rule: →±∞ x { } ± = ± lim ( ) ( ) f x g x A B 2. Sum/difference rule: →±∞ x { } = lim ( ). ( ) . f x g x A B 3. Product rule: →±∞ x ( ) f x A = lim ≠ ≠ 0, 0 g B 4. Quotient rule: B , →±∞ ( ) g x x ≤ ≤ 5. Comparison rule: If , then the ( ) ( ) ( ) f x h x g x ≤ ≤ limit lim ( ) lim ( ) h x A h x B exists and . →±∞ →±∞ x x 9/38 ------------------ Snezhana Gocheva-Ilieva, Plovdiv University ---------------------
4. Examples using limits laws at ±∞ Find the limits: − + 3 2 1 2 5 6 1 x x x − − lim 10 lim 3 lim a) , b) + , c) , − 3 →∞ 3 x x →∞ →−∞ 4 x x x x x + 3 2 1 x sin x lim lim d) x , e) − →∞ →∞ 1 x x x 1 2 1 2 − − = − − lim 10 lim lim lim10 Solution a). 3 3 →∞ →∞ →∞ →∞ x x x x x x x x 3 1 1 = − − = − − = − 3 lim 2lim lim10 0 2.0 10 10 . →∞ →∞ →∞ x x x x x 10/38 ------------------ Snezhana Gocheva-Ilieva, Plovdiv University ---------------------
6 x − 3 (5 ) x − 3 5 6 x x 3 x = lim lim − + 3 4 →∞ →∞ 3 4 x x x − + Solution b). 3 ( 3 ) x 3 x 6 1 x − − (5 ) (5 6lim ) − (5 0) 5 3 3 →∞ x x = = = − x lim − + 4 1 →∞ ( 3 0) 3 x − + − + ( 3 ) ( 3 4lim ) 3 3 →∞ x x x 1 + 2 1 x + 2 1 2 x x = lim lim Solution c). →−∞ →−∞ x x x x 1 1 + − + 1 ( ) 1 x x 2 2 1 x x = = = − + = − lim lim lim 1 1 2 →−∞ →−∞ →−∞ x x x x x x 11/38 ------------------ Snezhana Gocheva-Ilieva, Plovdiv University ---------------------
Solution d). We know that for all real x : − ≤ ≤ 1 sin 1 x . Now from the comparison rule: 1 sin 1 sin x x − ≤ ≤ − ≤ ≤ lim lim lim 0 lim 0 x , or . →∞ →∞ →∞ →∞ x x x x x x x sin x = lim 0. Therefore →∞ x x 12/38 ------------------ Snezhana Gocheva-Ilieva, Plovdiv University ---------------------
Solution e). + + 3 3 2 1 2 1 x x = lim lim − →∞ →∞ 1 x x x 1 − x 1 x ( ) 1 1 + 2 + 2lim lim 2 x x 2 x x x →∞ →∞ x x x x = = lim →∞ 1 1 x − − 1 1 lim x →∞ x x x ( ) + 2 2lim 0 x x = = = ∞ →∞ 2 x 2lim x x . − →∞ 1 0 x 13/38 ------------------ Snezhana Gocheva-Ilieva, Plovdiv University ---------------------
5. Remarkable limits at ±∞ x x 1 k + = + = k lim 1 , lim 1 e e →±∞ →∞ x x x x Examples. Find the limits: 3 x x 1 x + lim 1 lim a) + , b) →∞ →∞ 2 1 x x x x 1 x x 1 1 1 + = + = = 2 lim 1 lim 1 . e e Solution a). . →∞ →∞ 2 2 x x x x 14/38 ------------------ Snezhana Gocheva-Ilieva, Plovdiv University ---------------------
Solution b). 3 x 3 x 3 x 1 1 1 x − = = = = 3 lim lim e + 1 3 3 x →∞ →∞ 1 x x x e + 1 1 + . lim 1 x →∞ x x 3 x 3 x 3 x 1 1 1 x − = = = = 3 lim lim e + 1 3 3 x →∞ →∞ 1 x x x e + 1 1 + lim 1 x →∞ x x 15/38 ------------------ Snezhana Gocheva-Ilieva, Plovdiv University ---------------------
6. Infinite limits at x → a In many cases a function can grow or decrease infinitely when x approaches a finite number a . In fact this shows the behavior of the functions near a . Definition 4. We say that f ( x ) approaches infinity as x approaches a , and note = ∞ lim ( ) a f x → x If for every positive real number L there exist a corresponding number δ > 0 such that for all x satisfying < − < δ ( ) > ⇒ 0 x a L . f x 16/38 ------------------ Snezhana Gocheva-Ilieva, Plovdiv University ---------------------
Definition 5. We say that f ( x ) approaches minus infinity as x approaches a , and note = −∞ lim ( ) a f x → x If for every positive real number L there exist a corresponding number δ > 0 such that for all x satisfying < − < δ ( ) < − ⇒ 0 x a L . f x Remark. Remember, that the definitions 4-5 do not represent usual limits, these are only notations! 17/38 ------------------ Snezhana Gocheva-Ilieva, Plovdiv University ---------------------
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