<content> <p>lim</p> </content>

d0c02888-49fa-4e44-9297-e937796338b6 20 Валерий Очков Alvaro Diaz Falconi 4 5 false false 0 0 decimal

&[DATE] &[TIME] - &[FILENAME]

Euler spiral (clothoid) and its curvature

Δt 0.00001 :

t x π 2 / sqrt 2 π / sqrt t * FresnelC * :

t y π 2 / sqrt 2 π / sqrt t * FresnelS * :

t x' t Δt + x t x - Δt / :

t y' t Δt + y t y - Δt / :

t x'' t Δt + x' t x' - Δt / :

t y'' t Δt + y' t y' - Δt / :

t κ t y'' t x' * t y' t x'' * - abs t x' 2^t y' 2^+ (3^sqrt / :

10 κ 20

1000 κ 1999.9

100 κ 199.9999

10000 κ 19900.0855

T 06 0.1 range :

T x vectorize T y vectorize augment T T κ vectorize augment <content> <p>lim</p> </content>

E\032E\ vx\032E\ a\032E\ lim f\0324\ E\032E\ : x\0324\ vx\032E\ : XLIM Clear ans\0324\ : 14 mat δ'δ x\0324\ num2str concat str2num : f\0324\ x\0324\ v\0324\ r r\0324\ re(x\0324\ num2str): f\0324\ num2str concat : tmp\0324\ r\0324\ str2num : v\0324\ re 31 line : f\0324\ R\0324\ f\0324\ x\0324\ 0 r num2str 11 line : f\0324\ XLIM x\0324\ : diff num2str 0 ≡ (¬ a\032E\ num2str ∞ ≡ (a\032E\ num2str -∞ ≡ (| f\0324\ f\0324\ x\0324\ 1 x\0324\ / r : a\032E\ num2str 0 ≡ (¬ f\0324\ f\0324\ x\0324\ x\0324\ a\032E\ + r : 0 if if ans\0324\ f\0324\ R\0324\ : ans\0324\ str2num 21 line f\0324\ H\0324\ N\0324\ f\0324\ numden : lim.LH ¬ N\0324\ 1 el R\0324\ 0 ≡ (N\0324\ 2 el R\0324\ 0 ≡ (& (0 ≡ 1 try & ans\0324\ ans\0324\ f\0324\ R\0324\ : ans\0324\ str2num 21 line k\0324\ 1 MaxI.lim range ans\0324\ N\0324\ 1 el XLIM x\0324\ : k\0324\ diff N\0324\ 2 el XLIM x\0324\ : k\0324\ diff / R\0324\ : 0 try ans\0324\ ≡ continue break if 21 line for ans\0324\ 21 line try if 21 line : L\0324\ f\0324\ H\0324\ : (≡ L\0324\ f\0324\ ln H\0324\ : (≡ L\0324\ e f\0324\^H\0324\ : (≡ L\0324\ f\0324\ x\0324\ εl r : L\0324\ f\0324\ x\0324\ δ r : try L\0324\ IsDefined num2str 1 ≡ L\0324\ abs 0.01 εl 1 -^* > ∞ L\0324\ abs 100 εl * < 0 L\0324\ if if L\0324\ if 21 line L\0324\ str2num ln if 11 line e L\0324\ str2num^if 11 line L\0324\ str2num if 21 line try 21 line f\0324\ if 51 line :

Initialize

Replace functions (could equrep or at)

Convert to string for better work with try

Check if f depends of x

Convert ∞ to x -> 0

Convert a to x -> 0

Try to substitute, then f is continue

Check L'Hôpital conditions,
lim.LH helps

Try to apply LH rule some few times

Try to apply LH rule to f, ln(f) and exp(f)

Try to see if lim is 0 or ∞ numerically

Fails all attemps. return δx as infinitesimal.

LH rule of exp(f) works

LH rule of ln(f) works

LH rule works

f indep of x case

Defaults

MaxI.lim 8 :

εl 108 -^:

lim.LH 0 :

x.∞ t x t ∞ lim : 0.6267

t y t ∞ lim 0.6267

t x t y - t ∞ lim 0

t x t / t 0 lim 1

t y t / t 0 lim 0

εl 104 -^:

t κ t ∞ lim ∞

T x vectorize T κ vectorize augment T y vectorize T κ vectorize augment x 0 y * + x.∞ - 3 1 sys