Determinate And Indeterminate Limit Forms
Determinate and Indeterminate Limit FormsSome limits can be determined by inspection just by looking at the form of the limit – thesepredictable limit forms are called determinate. Other limits can’t be determined just by lookingat the form of the limit and can only be determined after additional work is done – theseunpredictable limit forms are called indeterminate.Determinate Limit Forms:Assuming that the functions involved in the limit are defined:1. The limit formsboundeda, for any number a, andresult in a limit of 0. 2. The limit form results in a limit of .3. The limit form results in a limit of 0.4. The limit form a , for 1 a 1, results in a limit of 0.5. The limit form a , for a 1, results in a limit of .6. The limit form a , for a 0 , results in a limit of .7. The limit form , for a 0 , results in a limit of 0.a8. The limit forms ,, and a , for a 0 , result in a limit of .0 a9. The limit forms ,, and a , for a 0 , result in a limit of .0 a10. The limit forms and result in a limit of .11. The limit forms , , and result in a limit of .12. The limit forms a 0 , for any number a, and bounded 0 result in a limit of 0.13. The limit forms a , for any number a, and bounded result in a limit of .14. The limit form a 0 , for a 0 , results in a limit of 1.
Indeterminate Limit Forms:1. 1 lim 1 1x x2x 0lim 1 lnxa a , for a 0xx lim 1 1x x2x lim 1 sinx x DNExx As you can see, this limit form can result in all limits from 0 to , and even DNE.2.00lim x 0x x2ax a , for any number ax 0 xlimlim x 0x x2x sin 1x xlim 2 DNE , lim DNEx 0 xx 0xAs you can see, this limit form can result in all limits from to , and even DNE.
3. x2 x xlimax a , for a 0x xlimx 0x x 2limx2lim x x2 x xsinx DNEx xlimAs you can see, this limit form can result in all limits from to , and even DNE.4. 0lim x 2 1x x lim x ax a , for any number ax lim x 2 1x x 1 lim 2 x xsinx DNEx x As you can see, this limit form can result in all limits from to , and even DNE.
5. 0 1 lim e x ln x 0x xlim 1a x 1x a , for 0 a 11lim x x 1x lim a x x a , for a 11x lim x x x 1x 1x xlim 3 sin x DNE x As you can see, this limit form can result in all limits from 0 to , and even DNE.6. 00lim x 1 3 ln x x 0 0ln alim x ln x a , for a 0x 0lim 12 x2 x 0 1 x lim x DNE , lim x sinx 0xx 0 2 1x1x DNExAs you can see, this limit form can result in all limits from 0 to , and even DNE.
7. lim x x 2 x lim x a x a , for any number ax lim x 2 x x lim x sin x x DNEx As you can see, this limit form can result in all limits from to , and even DNE.L’Hopital’s Rule Guidelines:Apply L’Hopital’s Rule toType of indeterminate form1. limf x 0f x or lim g x 0g x or limlimf x whatever g x f x g x lim 1 or lim 1 g x f x 2. lim f x g x 03. lim f x g x 1 ,lim f x f x g x g x 00 ,lim f x g x 0 ln f x g x lim orlim 1 g 1x ln f x But remember to exponentiateto get the original limit.4. lim f x g x f x 0 5. limg x whatever g 1x f 1x lim 1 g x f x lim1g x 1f x
Warning: If the zeros of g x accumulate at a, then it might be the case that limx aappears to exist, but limx af x g x f x f x . limg x x a g x Example:x cos x sin xx e x cos x sin x limsin xhas the form , and the limit doesn’t exist. However,x cos x sin x 1 sin 2 x cos 2 xlim lim sin xx x e sin x 1 sin 2 x cos2 x cos xesin x x cos x sin x . e x cos x sin x 2cos 2 x lim sin xx 2ecos 2 x cos xesin x x cos x sin x 2cos 2 xSince lim 0 , the discontinuities in the function 2 n 1 x 22esin x cos 2 x cos xesin x x cos x sin x 2cos 2 xcan be removed to yield the continuous function2esin x cos 2 x cos xesin x x cos x sin x 2cos x 2esin x cos x esin x x cos x sin x ;cos x 0 , and 0;cos x 0 limx 2esin x2cos x 0.cos x esin x x cos x sin x So a careless cancelation of cos x between the numerator and denominator would lead you tox cos x sin xbelieve that lim sin x 0 , when it actually doesn’t exist.x e x cos x sin x
As you can see, this limit form can result in all limits from 0 to f, and even DNE. 6. 0 0 1 3 ln 0 lim 0 x x x §· ¹ o ln ln 0 lim a x x xa o , for a!0 1 x 1 2 2 0 lim x xo ªº f «» ¼ 0 lim x x x DNE o , 2 1 1 0 lim sin x x x x x DNE o ªº ¼ As you can see, this lim it form can result in all limits from 0 to , and even DNE.
4. Design statically determinate and indeterminate axially loaded members including the case of thermal stresses. 5. Design statically determinate and indeterminate torsional loaded members including noncircular shafts. 6. Draw shear and bending moment diagrams for beams and shafts using the analytical and graphical
2. Analyse indeterminate trusses by approximate methods. 3. Analyse industrial frames and portals by approximate methods. 35.1 Introduction In module 2, force method of analysis is applied to solve indeterminate beams, trusses and frames. In modules 3 and 4, displacement based methods are discussed for the analysis of indeterminate structures.
Influence Lines for Statically Indeterminate Beams Reaction at A. 1 Scale Factor 1 E DE EE EE Vf f f §· ¹ Influence Lines for Statically Indeterminate Beams Shear at E. Influence Lines for Statically Indeterminate Beams Moment at E 1 Scale Factor 1
13 SOLUTION Externally stable, since the reactions are not concurrent or parallel.Since b 19, r 3, j 11, then b r 2j or 22 22. Therefore, the truss is statically determinate. By inspection the truss is internally stable. Externally stable.Since b 15, r 4, j 9, then b r 2j or 19 18. The truss is statically indeterminate to the first degree.By inspection the truss is .
Statically Indeterminate Problems Example: Consider a bar AB supported at both ends by fixed supports, with an axial force of 12 kN applied at C as illustrated. Find the reactions at the walls A C B 500 mm 400 mm 12 kN Solution: First, draw a free body diagram: Statically Indeterminate Problems and Problems Involving Two Materials – p. 5/30
Steps in Solving an Indeterminate Structure using the Force Method Determine degree of Indeterminacy Let n degree of indeterminacy (i.e. the structure is indeterminate to the nth degree) Define Primary Structure and the nRedundants Define the Primary Problem Solve for the n Relevant Deflections in Primary Problem Define the n Redundant Problems
C. Application of linear superposition to analyze indeterminate axially loaded members. Statically indeterminate structural systems are those where the number of unknown support reactions exceeds the number of global equilibrium equations. In such systems it is not possible to solve for the support
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