{"id":7440,"date":"2022-10-13T06:00:29","date_gmt":"2022-10-13T04:00:29","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=7440"},"modified":"2022-12-14T12:34:07","modified_gmt":"2022-12-14T10:34:07","slug":"aproximacion-al-limite-de-senx-x-cuando-x-tiende-a-cero","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/aproximacion-al-limite-de-senx-x-cuando-x-tiende-a-cero\/","title":{"rendered":"Aproximaci\u00f3n al l\u00edmite de sen(x)\/x cuando x tiende a cero"},"content":{"rendered":"<p>El limite de sen(x)\/x, cuando x tiende a cero, se puede calcular como el l\u00edmite de la sucesi\u00f3n sen(1\/n)\/(1\/n), cuando n tiende a infinito.<\/p>\n<p>Definir las funciones<\/p>\n<pre lang=\"text\">\n   aproxLimSeno :: Int -> [Double]\n   errorLimSeno :: Double -> Int\n<\/pre>\n<p>tales que<\/p>\n<ul>\n<li><code>aproxLimSeno n<\/code> es la lista de los <code>n<\/code> primeros t\u00e9rminos de la sucesi\u00f3n <code>sen(1\/m)\/(1\/m)<\/code>. Por ejemplo,<\/li>\n<\/ul>\n<pre lang=\"text\">\n     aproxLimSeno 1 == [0.8414709848078965]\n     aproxLimSeno 2 == [0.8414709848078965,0.958851077208406]\n<\/pre>\n<ul>\n<li><code>errorLimSeno x<\/code> es el menor n\u00famero de t\u00e9rminos de la sucesi\u00f3n <code>sen(1\/m)\/(1\/m)<\/code> necesarios para obtener su l\u00edmite con un error menor que <code>x<\/code>. Por ejemplo,<\/li>\n<\/ul>\n<pre lang=\"text\">\n     errorLimSeno 0.1     ==   2\n     errorLimSeno 0.01    ==   5\n     errorLimSeno 0.001   ==  13\n     errorLimSeno 0.0001  ==  41\n<\/pre>\n<p><b>Soluciones<\/b><\/p>\n<p>A continuaci\u00f3n se muestran las <a href=\"#haskell\">soluciones en Haskell<\/a> y las <a href=\"#python\">soluciones en Python<\/a>.<\/p>\n<p><a name=\"haskell\"><\/a><br \/>\n<b>Soluciones en Haskell<\/b><\/p>\n<pre lang=\"haskell\">\nimport Test.QuickCheck\n\n-- 1\u00aa definici\u00f3n de aproxLimSeno\n-- =============================\n\naproxLimSeno1 :: Int -> [Double]\naproxLimSeno1 k = [sin(1\/n)\/(1\/n) | n <- [1..k']]\n  where k' = fromIntegral k\n\n-- 2\u00aa definici\u00f3n de aproxLimSeno\n-- =============================\n\naproxLimSeno2 :: Int -> [Double]\naproxLimSeno2 0 = []\naproxLimSeno2 n = aproxLimSeno2 (n-1) ++ [sin(1\/n')\/(1\/n')]\n  where n' = fromIntegral n\n\n-- 3\u00aa definici\u00f3n de aproxLimSeno\n-- =============================\n\naproxLimSeno3 :: Int -> [Double]\naproxLimSeno3 = reverse . aux . fromIntegral\n  where aux 0 = []\n        aux n = sin(1\/n)\/(1\/n) : aux (n-1)\n\n-- 4\u00aa definici\u00f3n de aproxLimSeno\n-- =============================\n\naproxLimSeno4 :: Int -> [Double]\naproxLimSeno4 k = aux [] (fromIntegral k)\n  where aux xs 0 = xs\n        aux xs n = aux (sin(1\/n)\/(1\/n) : xs) (n-1)\n\n-- 5\u00aa definici\u00f3n de aproxLimSeno\n-- =============================\n\naproxLimSeno5 :: Int -> [Double]\naproxLimSeno5 k =  map (\\ n -> sin(1\/n)\/(1\/n)) [1..k']\n  where k' = fromIntegral k\n\n-- Comprobaci\u00f3n de equivalencia de aproxLimSeno\n-- ============================================\n\n-- La propiedad es\nprop_aproxLimSeno :: Positive Int -> Bool\nprop_aproxLimSeno (Positive k) =\n  all (== aproxLimSeno1 k)\n      [aproxLimSeno2 k,\n       aproxLimSeno3 k,\n       aproxLimSeno4 k,\n       aproxLimSeno5 k]\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_aproxLimSeno\n--    +++ OK, passed 100 tests.\n\n-- Comparaci\u00f3n de eficiencia de aproxLimSeno\n-- =========================================\n\n-- La comparaci\u00f3n es\n--    \u03bb> last (aproxLimSeno1 (2*10^4))\n--    0.9999999995833334\n--    (0.01 secs, 5,415,816 bytes)\n--    \u03bb> last (aproxLimSeno2 (2*10^4))\n--    0.9999999995833334\n--    (4.48 secs, 17,514,768,064 bytes)\n--    \u03bb> last (aproxLimSeno3 (2*10^4))\n--    0.9999999995833334\n--    (0.02 secs, 9,530,120 bytes)\n--    \u03bb> last (aproxLimSeno4 (2*10^4))\n--    0.9999999995833334\n--    (0.02 secs, 9,529,968 bytes)\n--    \u03bb> last (aproxLimSeno5 (2*10^4))\n--    0.9999999995833334\n--    (0.01 secs, 4,889,720 bytes)\n--\n--    \u03bb> last (aproxLimSeno1 (2*10^6))\n--    0.9999999999999583\n--    (0.46 secs, 480,569,808 bytes)\n--    \u03bb> last (aproxLimSeno3 (2*10^6))\n--    0.9999999999999583\n--    (1.96 secs, 896,569,992 bytes)\n--    \u03bb> last (aproxLimSeno4 (2*10^6))\n--    0.9999999999999583\n--    (1.93 secs, 896,570,048 bytes)\n--    \u03bb> last (aproxLimSeno5 (2*10^6))\n--    0.9999999999999583\n--    (0.05 secs, 432,569,800 bytes)\n--\n--    \u03bb> last (aproxLimSeno1 (10^7))\n--    0.9999999999999983\n--    (2.26 secs, 2,400,569,760 bytes)\n--    \u03bb> last (aproxLimSeno5 (10^7))\n--    0.9999999999999983\n--    (0.24 secs, 2,160,569,752 bytes)\n\n-- 1\u00aa definici\u00f3n de errorLimSeno\n-- =============================\n\nerrorLimSeno1 :: Double -> Int\nerrorLimSeno1 x =\n  round (head [m | m <- [1..], abs (1 - sin(1\/m)\/(1\/m)) < x])\n\n-- 2\u00aa definici\u00f3n de errorLimSeno\n-- =============================\n\nerrorLimSeno2 :: Double -> Int\nerrorLimSeno2 x = aux 1\n  where aux n | abs (1 - sin(1\/n)\/(1\/n)) < x = round n\n              | otherwise                    = aux (n+1)\n\n-- 3\u00aa definici\u00f3n de errorLimSeno\n-- =============================\n\nerrorLimSeno3 :: Double -> Int\nerrorLimSeno3 x =\n  round (head (dropWhile (\\ n -> abs (1 - sin(1\/n)\/(1\/n)) >= x) [1..]))\n\n-- Comprobaci\u00f3n de equivalencia de errorLimSeno\n-- ============================================\n\n-- La propiedad es\nprop_errorLimSeno :: Positive Double -> Bool\nprop_errorLimSeno (Positive x) =\n  all (== errorLimSeno1 x)\n      [errorLimSeno2 x,\n       errorLimSeno3 x]\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_errorLimSeno\n--    +++ OK, passed 100 tests.\n\n-- Comparaci\u00f3n de eficiencia de errorLimSeno\n-- =========================================\n\n-- La comparaci\u00f3n es\n--    \u03bb> errorLimSeno1 (10**(-12))\n--    408230\n--    (0.41 secs, 206,300,808 bytes)\n--    \u03bb> errorLimSeno2 (10**(-12))\n--    408230\n--    (0.46 secs, 225,895,672 bytes)\n--    \u03bb> errorLimSeno3 (10**(-12))\n--    408230\n--    (0.37 secs, 186,705,688 bytes)\n<\/pre>\n<p>El c\u00f3digo se encuentra en <a href=\"https:\/\/github.com\/jaalonso\/Exercitium\/blob\/main\/src\/Limite_del_seno.hs\">GitHub<\/a>.<\/p>\n<p><a name=\"python\"><\/a><br \/>\n<b>Soluciones en Python<\/b><\/p>\n<pre lang=\"python\">\nfrom itertools import dropwhile, islice\nfrom math import sin\nfrom sys import setrecursionlimit\nfrom timeit import Timer, default_timer\nfrom typing import Iterator\n\nfrom hypothesis import given\nfrom hypothesis import strategies as st\n\nsetrecursionlimit(10**6)\n\n# 1\u00aa definici\u00f3n de aproxLimSeno\n# =============================\n\ndef aproxLimSeno1(k: int) -> list[float]:\n    return [sin(1\/n)\/(1\/n) for n in range(1, k + 1)]\n\n# 2\u00aa definici\u00f3n de aproxLimSeno\n# =============================\n\ndef aproxLimSeno2(n: int) -> list[float]:\n    if n == 0:\n        return []\n    return aproxLimSeno2(n - 1) + [sin(1\/n)\/(1\/n)]\n\n# 3\u00aa definici\u00f3n de aproxLimSeno\n# =============================\n\ndef aproxLimSeno3(n: int) -> list[float]:\n    def aux(n: int) -> list[float]:\n        if n == 0:\n            return []\n        return [sin(1\/n)\/(1\/n)] + aux(n - 1)\n\n    return list(reversed(aux(n)))\n\n# 4\u00aa definici\u00f3n de aproxLimSeno\n# =============================\n\ndef aproxLimSeno4(n: int) -> list[float]:\n    def aux(xs: list[float], n: int) -> list[float]:\n        if n == 0:\n            return xs\n        return aux([sin(1\/n)\/(1\/n)] + xs, n - 1)\n\n    return aux([], n)\n\n# 5\u00aa definici\u00f3n de aproxLimSeno\n# =============================\n\ndef aproxLimSeno5(n: int) -> list[float]:\n    return list(map((lambda k: sin(1\/k)\/(1\/k)), range(1, n+1)))\n\n# 6\u00aa definici\u00f3n de aproxLimSeno\n# =============================\n\ndef aproxLimSeno6(n: int) -> list[float]:\n    r = []\n    for k in range(1, n+1):\n        r.append(sin(1\/k)\/(1\/k))\n    return r\n\n# Comprobaci\u00f3n de equivalencia de aproxLimSeno\n# ============================================\n\n# La propiedad es\n@given(st.integers(min_value=1, max_value=100))\ndef test_aproxLimSeno(n: int) -> None:\n    r = aproxLimSeno1(n)\n    assert aproxLimSeno2(n) == r\n    assert aproxLimSeno3(n) == r\n    assert aproxLimSeno4(n) == r\n    assert aproxLimSeno5(n) == r\n    assert aproxLimSeno6(n) == r\n\n# La comprobaci\u00f3n es\n#    src> poetry run pytest -q limite_del_seno.py\n#    1 passed in 0.60s\n\n# Comparaci\u00f3n de eficiencia de aproxLimSeno\n# =========================================\n\ndef tiempo(e: str) -> None:\n    \"\"\"Tiempo (en segundos) de evaluar la expresi\u00f3n e.\"\"\"\n    t = Timer(e, \"\", default_timer, globals()).timeit(1)\n    print(f\"{t:0.2f} segundos\")\n\n# La comparaci\u00f3n es\n#    >>> tiempo('aproxLimSeno1(3*10**5)')\n#    0.03 segundos\n#    >>> tiempo('aproxLimSeno2(3*10**5)')\n#    Process Python violaci\u00f3n de segmento (core dumped)\n#    >>> tiempo('aproxLimSeno3(3*10**5)')\n#    Process Python violaci\u00f3n de segmento (core dumped)\n#    >>> tiempo('aproxLimSeno4(3*10**5)')\n#    Process Python violaci\u00f3n de segmento (core dumped)\n#    >>> tiempo('aproxLimSeno5(3*10**5)')\n#    0.04 segundos\n#    >>> tiempo('aproxLimSeno6(3*10**5)')\n#    0.07 segundos\n#\n#    >>> tiempo('aproxLimSeno1(10**7)')\n#    1.29 segundos\n#    >>> tiempo('aproxLimSeno5(10**7)')\n#    1.40 segundos\n#    >>> tiempo('aproxLimSeno6(10**7)')\n#    1.45 segundos\n\n# 1\u00aa definici\u00f3n de errorLimSeno\n# ============================\n\n# naturales es el generador de los n\u00fameros naturales positivos, Por\n# ejemplo,\n#    >>> list(islice(naturales(), 5))\n#    [1, 2, 3, 4, 5]\ndef naturales() -> Iterator[int]:\n    i = 1\n    while True:\n        yield i\n        i += 1\n\ndef errorLimSeno1(x: float) -> int:\n    return list(islice((n for n in naturales()\n                        if abs(1 - sin(1\/n)\/(1\/n)) < x), 1))[0]\n\n# 2\u00aa definici\u00f3n de errorLimSeno\n# ============================\n\ndef errorLimSeno2(x: float) -> int:\n    def aux(n: int) -> int:\n        if abs(1 - sin(1\/n)\/(1\/n)) < x:\n            return n\n        return aux(n + 1)\n\n    return aux(1)\n\n# 3\u00aa definici\u00f3n de errorLimSeno\n# ============================\n\ndef errorLimSeno3(x: float) -> int:\n    return list(islice(dropwhile(lambda n: abs(1 - sin(1\/n)\/(1\/n)) >= x,\n                                 naturales()),\n                       1))[0]\n\n\n# Comprobaci\u00f3n de equivalencia de errorLimSeno\n# ============================================\n\n@given(st.integers(min_value=1, max_value=100))\ndef test_errorLimSeno(n: int) -> None:\n    r = errorLimSeno1(n)\n    assert errorLimSeno2(n) == r\n    assert errorLimSeno3(n) == r\n\n# La comprobaci\u00f3n es\n#    src> poetry run pytest -q limite_del_seno.py\n#    2 passed in 0.60s\n\n# Comparaci\u00f3n de eficiencia de errorLimSeno\n# =========================================\n\n# La comparaci\u00f3n es\n#    >>> tiempo('errorLimSeno1(10**(-12))')\n#    0.07 segundos\n#    >>> tiempo('errorLimSeno2(10**(-12))')\n#    Process Python violaci\u00f3n de segmento (core dumped)\n#    >>> tiempo('errorLimSeno3(10**(-12))')\n#    0.10 segundos\n<\/pre>\n<p>El c\u00f3digo se encuentra en <a href=\"https:\/\/github.com\/jaalonso\/Exercitium-Python\/blob\/main\/src\/limite_del_seno.py\">GitHub<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>El limite de sen(x)\/x, cuando x tiende a cero, se puede calcular como el l\u00edmite de la sucesi\u00f3n sen(1\/n)\/(1\/n), cuando n tiende a infinito. Definir las funciones aproxLimSeno :: Int -> [Double] errorLimSeno :: Double -> Int tales que aproxLimSeno n es la lista de los n primeros t\u00e9rminos de la sucesi\u00f3n sen(1\/m)\/(1\/m). Por ejemplo,&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_kad_post_transparent":"","_kad_post_title":"","_kad_post_layout":"","_kad_post_sidebar_id":"","_kad_post_content_style":"","_kad_post_vertical_padding":"","_kad_post_feature":"","_kad_post_feature_position":"","_kad_post_header":false,"_kad_post_footer":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"footnotes":"","_jetpack_memberships_contains_paid_content":false},"categories":[581],"tags":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7440"}],"collection":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/comments?post=7440"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7440\/revisions"}],"predecessor-version":[{"id":7672,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7440\/revisions\/7672"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=7440"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=7440"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=7440"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}