ó ¡¼™\c@ s?ddlmZmZddlmZmZmZmZmZm Z m Z ddl m Z ddl mZddlmZddlmZddlmZddlmZdd lmZdd lmZmZdd lmZdd lmZdd l m!Z!ddl"m"Z"dd„Z#d„Z$defd„ƒYZ%dS(iÿÿÿÿ(tprint_functiontdivision(tStSymboltAddtsympifytExprt PoleErrortMul(t string_types(t factor_terms(t GoldenRatio(tDummy(t factorial(t fibonacci(tgamma(tPolynomialErrortfactor(tOrder(tratsimp(ttogetheri(tgruntzt+cC s |dkr}t|||ddƒjdtƒ}t|||ddƒjdtƒ}||krd|Std||fƒ‚nt||||ƒjdtƒSdS(säComputes the limit of ``e(z)`` at the point ``z0``. Parameters ========== e : expression, the limit of which is to be taken z : symbol representing the variable in the limit. Other symbols are treated as constants. Multivariate limits are not supported. z0 : the value toward which ``z`` tends. Can be any expression, including ``oo`` and ``-oo``. dir : string, optional (default: "+") The limit is bi-directional if ``dir="+-"``, from the right (z->z0+) if ``dir="+"``, and from the left (z->z0-) if ``dir="-"``. For infinite ``z0`` (``oo`` or ``-oo``), the ``dir`` argument is determined from the direction of the infinity (i.e., ``dir="-"`` for ``oo``). Examples ======== >>> from sympy import limit, sin, Symbol, oo >>> from sympy.abc import x >>> limit(sin(x)/x, x, 0) 1 >>> limit(1/x, x, 0) # default dir='+' oo >>> limit(1/x, x, 0, dir="-") -oo >>> limit(1/x, x, 0, dir='+-') Traceback (most recent call last): ... ValueError: The limit does not exist since left hand limit = -oo and right hand limit = oo >>> limit(1/x, x, oo) 0 Notes ===== First we try some heuristics for easy and frequent cases like "x", "1/x", "x**2" and similar, so that it's fast. For all other cases, we use the Gruntz algorithm (see the gruntz() function). See Also ======== limit_seq : returns the limit of a sequence. s+-tdirt-tdeepRsMThe limit does not exist since left hand limit = %s and right hand limit = %sN(tLimittdoittFalset ValueError(tetztz0Rtllimtrlim((s2lib/python2.7/site-packages/sympy/series/limits.pytlimits6 $$ c sddlm‰d}t|ƒtjkr~t|j|d|ƒ|tj|tjkr_dndƒ}t |t ƒrdSn|j s¢|j s¢|j s¢|jrg}xû|jD]ð}t||||ƒ}|jtjƒro|jdkrot |tƒrkt|ƒ}t |tƒs't|ƒ}nt |tƒsEt|ƒ}nt |tƒrgt||||ƒSdSdSt |t ƒr‚dS|tjkr•dS|j|ƒq²W|r|j|Œ}|tjkr¢|j r¢t‡fd†|Dƒƒr¢g} g} xUtt|ƒƒD]A} t || ˆƒr;| j|| ƒq| j|j| ƒqWt| ƒdkr¢t| Œjƒ} t| |||ƒ}|t| Œ}q¢n|tjkr yt|ƒ} Wnt k rÕdSX| tjksñ| |krõdSt| |||ƒSqn|S( s+Computes the limit of an expression term-wise. Parameters are the same as for the ``limit`` function. Works with the arguments of expression ``e`` one by one, computing the limit of each and then combining the results. This approach works only for simple limits, but it is fast. iÿÿÿÿ(t AccumBoundsiRRNc3 s|]}t|ˆƒVqdS(N(t isinstance(t.0trr(R$(s2lib/python2.7/site-packages/sympy/series/limits.pys zsi(!tsympy.calculus.utilR$tNonetabsRtInfinityR#tsubstZeroR%Rtis_Multis_Addtis_Powt is_Functiontargsthast is_finiteRR RRRt heuristicstNaNtappendtfunctanytrangetlentsimplifyRR(RRR Rtrvtrtatltmtr2te2tiite3trat_e((R$s2lib/python2.7/site-packages/sympy/series/limits.pyR5Us`=$! 4 RcB s2eZdZdd„Zed„ƒZd„ZRS(sRepresents an unevaluated limit. Examples ======== >>> from sympy import Limit, sin, Symbol >>> from sympy.abc import x >>> Limit(sin(x)/x, x, 0) Limit(sin(x)/x, x, 0) >>> Limit(1/x, x, 0, dir="-") Limit(1/x, x, 0, dir='-') RcC sçt|ƒ}t|ƒ}t|ƒ}|tjkr<d}n|tjkrTd}nt|tƒrrt|ƒ}n(t|tƒsštdt|ƒƒ‚nt |ƒdkr¿t d|ƒ‚nt j |ƒ}||||f|_ |S(NRRs6direction must be of type basestring or Symbol, not %ss+-s1direction must be one of '+', '-' or '+-', not %s(RRs+-(RRR+tNegativeInfinityR%R Rt TypeErrorttypetstrRRt__new__t_args(tclsRRR Rtobj((s2lib/python2.7/site-packages/sympy/series/limits.pyRK¢s$      cC sH|jd}|j}|j|jdjƒ|j|jdjƒ|S(Niii(R2t free_symbolstdifference_updatetupdate(tselfRtisyms((s2lib/python2.7/site-packages/sympy/series/limits.pyROºs   c  s’ddlm}ddlm}|j\}‰}}|tjkrStdƒ‚n|jdt ƒr•|j |}ˆj |‰|j |}n|ˆkr¥|S|j ˆƒs¸|S|j rÜ|j t|gtƒ}n|jrït|ƒtjkrït|ƒ}|j ttƒ}‡fd†‰t‡fd†|jƒDƒƒrìtdt ƒ}|tjkr€|jˆd|ƒ}n|jˆd |ƒ}y<t|j|ƒ|tjd ƒ} t| tƒrÍ|S| SWqét k råqéXqìqïn|j!rt"t|j#ˆ|ƒ|jd ŒSy4t$|ˆ||ƒ} | tj%krNt&ƒ‚nWn<t&t fk rt'|ˆ||ƒ} | d krŽ|SnX| S( sPEvaluates the limit. Parameters ========== deep : bool, optional (default: True) Invoke the ``doit`` method of the expressions involved before taking the limit. hints : optional keyword arguments To be passed to ``doit`` methods; only used if deep is True. iÿÿÿÿ(t limit_seq(tRisingFactorials.Limits at complex infinity are not implementedRc s2ˆ|jko1t‡fd†tj|ƒDƒƒS(Nc3 sC|]9}|jˆƒp:t‡fd†tj|ƒDƒƒVqdS(c3 s-|]#}ˆ|jko$|jˆƒVqdS(N(ROt is_polynomial(R&RA(R(s2lib/python2.7/site-packages/sympy/series/limits.pys ñsN(RVR9Rt make_args(R&R?(R(s2lib/python2.7/site-packages/sympy/series/limits.pys ðs(ROR9RRW(tw(R(s2lib/python2.7/site-packages/sympy/series/limits.pytïsc3 s|]}ˆ|ƒVqdS(N((R&RX(tok(s2lib/python2.7/site-packages/sympy/series/limits.pys ôstpositiveiRN()tsympy.series.limitseqRTtsympy.functionsRUR2RtComplexInfinitytNotImplementedErrortgettTrueRR3t is_positivetrewriteR RR.R*R+R RR talltas_numer_denomR RGR,R#tas_leading_termR-R%RRtis_OrderRtexprRR6RR5R)( RRthintsRTRURR RtutinveR>((RZRs2lib/python2.7/site-packages/sympy/series/limits.pyRÃsV     "!   # (t__name__t __module__t__doc__RKtpropertyROR(((s2lib/python2.7/site-packages/sympy/series/limits.pyR“s   N(&t __future__RRt sympy.coreRRRRRRRtsympy.core.compatibilityR tsympy.core.exprtoolsR tsympy.core.numbersR tsympy.core.symbolR t(sympy.functions.combinatorial.factorialsR t%sympy.functions.combinatorial.numbersRt'sympy.functions.special.gamma_functionsRt sympy.polysRRtsympy.series.orderRtsympy.simplify.ratsimpRtsympy.simplify.simplifyRRR#R5R(((s2lib/python2.7/site-packages/sympy/series/limits.pyts 4 D >