VIEWS: 4 PAGES: 8 CATEGORY: Personal Finance POSTED ON: 9/28/2010
Limits and the Infinity Limits at the Infinity How to Compute Limits at the Infinity The infinity as a Limit Index FAQ Limit at the Infinity (1) The definition of limits has to be modified for the case where the variable approaches or - . Definition 1 A function f has a finite limit L as x if the values of the function f get arbitrarily close to the number L as x grows. Notation lim f x L x The same definition in a rigorous way. Definition 2 lim f x L 0 : M such that x x M f x L . Index Mika Seppälä: Limits and the Infinity FAQ Limit at the Infinity (2) Definition 2 lim f x L 0 : M such that x x M f x L . sin x Example lim 0. x x Proof Let 0 be given. sin x sin x 1 1 0 if x M. x x x Index Mika Seppälä: Limits and the Infinity FAQ Limit at the Infinity (3) Definition 1 A function f has a finite limit L as x- if the values of the function f get arbitrarily close to the number L as x gets smaller. Notation lim f x L x The same definition in a rigorous way. Definition 2 lim f x L 0 : M such that x x M f x L . Index Mika Seppälä: Limits and the Infinity FAQ How to Compute Limits at the Infinity 2x 3 5 x 2 x 1 Example lim x x3 1 To compute limits of the above type, the main method is to rewrite the expression so that, after substitution x = , the expression can be simplified to a number using the rule c / = 0, where c is any finite number. In the above example, direct substitution does not work because it leads to the undefined expression / . 5 1 1 2 2 3 After substitution 2x 5 x x 1 3 2 x x x . Solution Rewrite x=, all the circled x3 1 1 1 3 terms become 0. x 5 1 1 2 2 3 Substitute x to conclude x x x 2. x 1 1 3 x Index Mika Seppälä: Limits and the Infinity FAQ The Infinity as a Limit (1) Definition 1 Let x0 a finite number. A function f has the infinity as its limit as xx0 if the values of the function f grow arbitrarily large as x gets close to x0. Notation lim f x x x0 The same definition in a rigorous way. Definition 2 lim f x M : 0 such that x x0 0 x x0 f x M. Index Mika Seppälä: Limits and the Infinity FAQ The Infinity as a Limit (2) Definition 2 lim f x M : 0 such that x x0 0 x x0 f x M. 1 Example lim 2 x 0 x Proof Let M 0 be given. 1 1 1 2 M x 2 x 0x x M M Index Mika Seppälä: Limits and the Infinity FAQ The Infinity as a Limit (3) Problem Following the outline of the previous definitions define: lim f x . x Solution and lim f x M : m such that x Definition x m f x M. Problem Following the outline of the previous definitions define: lim f x . x Solution and lim f x M : m such that x Definition x m f x M. Index Mika Seppälä: Limits and the Infinity FAQ