a 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, 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='+-') zoo >>> 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. F)deep)Limitdoit)ezz0dirr#c/Users/vegardjervell/Documents/master/model/venv/lib/python3.9/site-packages/sympy/series/limits.pylimits3r%cCs<d}t|tjurNt||d||tj|tjur6dnd}t|trJdSn|jsh|j sh|j sh|j r8g}|j D]}t||||}| tjr|jdurt|trt|}t|tst|}t|tst|}t|trt||||SdSdSt|tr dS|tjurdS||qr|r8|j|}|tjur|jrtdd|Drg} g} tt|D]6} t|| tr| || n| |j | qtt| dkrt| } t| |||}|t| }|tjur8z t|} WntyYdS0| tjus&| |kr*dSt| |||S|S)a+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. Nrr-css|]}t|tVqdSN) isinstancer).0rrr#r#r$ jzheuristics..r)absrInfinityr%subsZeror(ris_MulZis_Addis_PowZ is_Functionargshas is_finiterr r rr heuristicsNaNappendfuncanyrangelenrZsimplifyrr)rr r!r"rvralmZr2e2iiZe3Zrat_er#r#r$r6Fs^*          (    r6c@s6eZdZdZd ddZeddZddZd d Zd S) rzRepresents an unevaluated limit. Examples ======== >>> from sympy import Limit, sin >>> from sympy.abc import x >>> Limit(sin(x)/x, x, 0) Limit(sin(x)/x, x, 0) >>> Limit(1/x, x, 0, dir="-") Limit(1/x, x, 0, dir='-') rcCst|}t|}t|}|tjur(d}n|tjur6d}||rPtd||ft|trdt|}nt|ts~t dt |t|dvrt d|t |}||||f|_|S)Nr&rz@Limits approaching a variable point are not supported (%s -> %s)z6direction must be of type basestring or Symbol, not %s)rr&+-z1direction must be one of '+', '-' or '+-', not %s)rrr.NegativeInfinityr4NotImplementedErrorr(strr TypeErrortype ValueErrorr__new___args)clsrr r!r"objr#r#r$rKs0        z Limit.__new__cCs8|jd}|j}||jdj||jdj|S)Nrr)r3 free_symbolsdifference_updateupdate)selfrZisymsr#r#r$rPs  zLimit.free_symbolsc Cs|j\}}}}|j|j}}||sBt|t|||}t|St|||}t|||} | tjur|tjtj fvrt||d||}t|S| tj ur|tjurtj SdS)Nr) r3baserr4r%rrOner.rEComplexInfinity) rSr_r r!b1e1resZex_limZbase_limr#r#r$pow_heuristicss    zLimit.pow_heuristicsc sD|j\}tjur td|ddr\|jfi|}jfi|jfi||krhS|sv|StjurtjS|tjtj tjtjr|S|j rt t |j g|jddRSd}tdkrd}ntdkrd }fd d |trt|}|}|rttjurV|d}| }n|}z|j|d \}}WntyYnd0|dkrtjS|dkr|S|dkst|d@stjt|S|d krtj t|StjSttjur,|jrt|}|d}| }n|}z|j|d \}}WnFtttfyt|}|jr||}|dur|YSYn0|tjtj tjr|S|sl|jrtjS|dkr|S|j rl|j!r,|dks|j"r tjt|S|d kr$tj t|StjSn@|dkrDtjt|S|d krftjt|tj#|StjSj$r|%t&t'}d}zvtd krt(|d}t(|d}||krtd||fnt(|}|tjus|tjurtWnDttfy>|durt)|}|dur:|YSYn0|S)aPEvaluates 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. z.Limits at complex infinity are not implementedrTrNrrr&cs|js |Stfdd|jD}||jkr6|j|}t|t}t|t}|sR|rt|jd}|jrtd|jd}|jr|dkdkr|r|jd St j S|dkdkr|r|jdSt j S|S)Nc3s|]}|VqdSr'r#)r)arg) set_signsr#r$r+r,z0Limit.doit..set_signs..rrT) r3tupler9r(r rr%is_zeroZis_extended_realr NegativeOnerU)exprZnewargsZabs_flagZ sign_flagsigr"r^r r!r#r$r^s"      zLimit.doit..set_signs)cdirrDzMThe limit does not exist since left hand limit = %s and right hand limit = %s)*r3rrVrFgetrr4r7r.rEZis_Orderrr%rbrGr rZis_meromorphicr-r/ZleadtermrJr0intrr1r rrr2r[Z is_positiveZ is_negative is_integerZis_evenraZis_extended_positiveZrewriter rrr6) rShintsrreZneweZcoeffexr>r@r#rdr$rs    $                  z Limit.doitN)r) __name__ __module__ __qualname____doc__rKpropertyrPr[rr#r#r#r$rs   rN)r)(Z!sympy.calculus.accumulationboundsrZ sympy.corerrrrrrr Zsympy.core.exprtoolsr Zsympy.core.numbersr Z(sympy.functions.combinatorial.factorialsr Z$sympy.functions.elementary.complexesr rZ&sympy.functions.elementary.exponentialrrZ'sympy.functions.special.gamma_functionsrZ sympy.polysrrZsympy.series.orderrZsympy.simplify.powsimprZsympy.simplify.ratsimprZsympy.simplify.simplifyrrrr%r6rr#r#r#r$s $         6=