<content> lim </content>

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Limits

4 appVersion 0.99.7822.147

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 :

all MaximaTakeover diff(), int(), lim(), sum(), det() handled by Maxima

See how Maxima can handle the examples. The cases with differences to Alvaro's original version are highlighted with boxes.

<content> lim examples </content>

Extended real number line

SMath rules assumes a > 1 finite number, and have some bugs.

a ∞ + ∞

a ∞ - ∞ -

But this two cases are wrong:

a ∞ * ∞

a ∞ / 0

0.5 ∞^∞

∞ a^∞

∞ a -^0

0.5 ∞ -^0

a ∞^∞

a ∞ -^0

Trivial cases

x f x a lim x f x a lim

x sin x / x a lim a sin a /

a x 2^* x + x a lim a 1 a 2^+ (*

5 x * x ∞ lim ∞

4 x 2^* 5 x * - x ∞ lim ∞

a x 2^* b x * - x ∞ lim a is assumed to be positive. ∞ 21 sys

Quotient of polinomials

x 2^25 - x 2^2 x * + 15 - / x 5 - lim 54 /

5 x * 3 + 3 x 2^* 2 x * - / x ∞ lim 0

x 2^5 x * - 6 + x 3 - / x 3 lim 1

5 x 2^* 3 + 3 x 2^* 2 x * - / x ∞ lim 53 /

x 2^4 x * + 12 - x 2^2 x * - / x 2 lim 4

5 x 3^* 3 + 3 x 2^* 2 x * - / x ∞ lim ∞

With another unknow sometimes can get the result, ando sometimes returns the infinitesimal form

x 3^8 x 2^* - a x * + 18 - 2 x 2^* 12 x * - b + / x 3 lim 321 - a + (* 18 - b + /

x 3^8 x 2^* - 21 x * + 18 - 2 x 2^* 12 x * - 18 + / x 3 lim 12 /

2 x 3 - (2^* x / x 0 lim ∞

2 x 3 a * - (2^* x / x 0 lim a is assumed to be positive. ∞ 21 sys

aa 0 > assume Maxima aa 0 > 11 sys

2 x 3 aa * - (2^* x / x 0 lim ∞

Infinity and numerical limits

If there are not other variables, and L'Hôpital rule fails, lim try to simplify the expression to zero or infinity in a numerical aproximated way, so, it can result in a wrong value. But I can't get any example of that.

3 x 2^* 2 x * - 5 x * 3 + / x ∞ lim ∞

1 x 2 - (3^/ x 2 lim ∞

2 x 3 - (2^* x / x 0 lim ∞

x 2^2 x * - x 3 + / x ∞ lim ∞

x x ln / x ∞ lim ∞

x ln x / x 0 lim ∞

a x * ln a x * / x 0 lim a is assumed to be positive. ∞ 21 sys

Potential, exponential and logarithmic functions

1 a x * + b x * - nthroot x 0 lim 1 e b nthroot a^/

x x a + / b x / nthroot x ∞ lim 1 e b nthroot a^/

1 a x * + x b / nthroot x 0 lim e a b *^

1 a x / + (b x *^x ∞ lim e a b *^

x 1 + a x * nthroot x 0 lim e a nthroot

x 1 + x / a x / nthroot x ∞ lim e a nthroot

x 1 + x a / nthroot x 0 lim e a^

x 1 + x / (a x *^x ∞ lim e a^

e a x *^1 - x / x 0 lim a

x e x a / nthroot 1 - (* x ∞ lim a

1 a x * + ln x / x 0 lim a

x 1 a x / + ln x * * x ∞ lim a is assumed to be positive. ∞ 21 sys

x ln x 2^/ x ∞ lim 0

x 2^x ln / x ∞ lim ∞

1 x / ln x 2^/ x ∞ lim 0

x 2^1 x / ln * x ∞ lim ∞ -

Trigonometric functions

a x * sin b x * / x 0 lim a b /

1 a x * cos - b x 2^* / x 0 lim a 2^2 b * /

a x * tan b x * / x 0 lim a b /

x 2^sin x 2^/ x 0 lim 1

1 x 2^cos - x 2^/ x 0 lim 0

1 a x * cos - b x * / x 0 lim 0

x 2^1 x / cos * x 0 lim 0

x x sin * x 0 lim 0

x x sin * x ∞ lim und

Gallery

1 x 3^/ 2 x cos + 3 / (x^1 - (* x 0 lim 16 / -

2 x * sin x sin / x π lim 2 -

x tan x - x 3^/ x 0 lim 13 /

15 x * sin + (1 x tan /^x 0 lim e 5^

x 3^x 2^- 3 x * + sqrt x 3^sqrt x 2^sqrt - 3 x * sqrt + / x ∞ lim 1

x x 2^1 + sqrt / x ∞ lim 1

Indeterminate forms

All error messages are ok, except for 0/0:

00 / #

lastError Division by zero.

∞ ∞ - #

lastError Uncertainty.

x sin a x * - x 3^/ x 0 lim a is assumed to be positive. a-1 is assumed to be positive. ∞ - 31 sys

4 x * x 2^ln - x ∞ lim ∞

∞ ∞ / #

lastError Uncertainty.

∞ 0^#

lastError Uncertainty.

x ln x 3^/ x ∞ lim 0

x x nthroot x ∞ lim 1

Not a big news

0 ∞ * #

lastError Uncertainty.

1 ∞^#

lastError Uncertainty.

x x ln * x 0 lim 0

11 x / + (x^x ∞ lim e

00^#

lastError Uncertainty.

x x x ln *^x 0 lim 1

Derivatives

This is more or less obvious just becaus lim is based on diff.

x f x a x * cos * :

x h + f x f - h / h 0 lim a x * a x * sin * - a x * cos +

x f a x * b + sqrt :

x h + f x f - h / h 0 lim a 2 b a x * + sqrt * /

Using the infinitesimal δx

We can study the behaviour of the limit when it returns δ's

y t 2^a 2 / + abs t nthroot :

y t 0 lim 2-abs(a) is assumed to be positive. 1 is assumed to be positive. ∞ 31 sys

Known issues

If a function gives an error with diff, it will be given an error with lim too.

xo 4.5 :

xo Gamma xo diff diff: second argument must be a variable; found 9/2
 -- an error. To debug this try: debugmode(true);
(%i89)

x 3 x * 4 + sqrt - 4 x - / x 4 lim 58 / -

x 1 x - / x 1 plus limit Maxima ∞ -

right/left side limits used to be controlled by adding +0 or -0 to the point (e.g. \[REGION[9077474fc9eb44a887a2bb4802625951]]\). This is suppressed now by mandatory pre-processing.

40 +

x 1 x - / x 1 minus limit Maxima ∞

you still can use the four-argument form of limit in Maxima.

y 1 a e x -^1 - (* + x - nthroot :

y x 0 lim e a^

The bug
is here:

y ln e x^a e x^- 1 + (* + ln x + x / -

correct
value:

y ln 1 a e x -^1 - (* + ln x / - ≡

The code can't handle very well complex expressions or some other expressions, returning numerical approximations instead exact values

u 1 x + (1 x /^:

v x 2^12 - x 2^2 - / :

z u i v * + :

z x 0 lim e 6 i * +

even

u x 0 lim e

v x 0 lim 6

x sin x / x 3 lim 3 sin 3 /

even

x sin x / x a lim a sin a /

But at least you can evaluate both numerically

x sin x / x 3 lim 0.047

z x 0 lim 2.7183 6 i * +

There are an option for control the numerical answers, εl value. For example, here the code can try to get a numeric answer, but there are an (not visible for the user) overflow error.

x e x -^* x ∞ lim 0

x e x nthroot * x 0 lim und

εl 101 -^:

x e x -^* x ∞ lim 0

x e x nthroot * x 0 lim und

εl 108 -^:

The code do not recognice directions, which can introduce some issues. For example, let

y.1 132 x nthroot + / :

y.1 x 0 lim und

Both are function only of x, for get a closed result change εl value

y.1 x 0 minus limit Maxima 13 /

y.1 x 0 plus limit Maxima 0

y.2 12 x nthroot + 32 x nthroot + / :

y.2 x 0 lim und

y.2 x 0 minus limit Maxima 13 /

y.1 x 0 plus limit Maxima 0

y.1 x 0 \002D\^lim 13 / ≡

y.1 x 0 lim und

but

y.1 x 0 \002B\^lim 0 ≡

and

Nice idea for specification of left and right side limits. That could be implemented in the Maxima Plugin, perhaps using a custom translation

y.1 y.2 2 1 sys

y x x cos x sin * + e x sin^x x cos x sin * + (* / :

y x ∞ lim ind

Alvaro

(Martin Kraska)