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Calculus`Limit`

This package provides an enhancement to the built-in Limit. It allows you to find limits of expressions that contain a wide class of elementary and special functions. This package cannot handle expressions containing hypergeometric, elliptic and certain other special functions.
The syntax and semantics of the enhanced Limit are the same as those of the built-in Limit.


Finding limits.

  • This loads the package.
  • In[1]:= <<Calculus`Limit`

  • The package can handle the limit of a combination of expressions having head Power.
  • In[2]:= Limit[E^x^x - E^x^(2 x), x->Infinity]

    Out[2]=

  • Here the enhanced Limit is applied to a combination of logarithmic functions.
  • In[3]:= Limit[(Log[Cot[-x/Log[x]]] - Log[1/x]) /
    Log[Log[x]], x->0]

    Out[3]=

  • Like the built-in Limit, the enhanced Limit recognizes the Direction option indicating a directional limit.
  • In[4]:= Limit[Exp[Tan[x]/Log[Cos[x]] ],
    x->Pi/2, Direction->-1]

    Out[4]=

  • This gives the limit of an expression involving the exponential integral function.
  • In[5]:= Limit[E^x ExpIntegralE[2, ArcTan[E^x] - Pi/2] -
    E^x - x, x->Infinity]

    Out[5]=

  • The enhanced Limit can handle a combination of logarithmic and polylogarithmic functions.
  • In[6]:= Limit[PolyLog[2, x] + Log[x]^2/2 +
    I Pi Log[x], x ->Infinity]

    Out[6]=

  • This gives the limit of an expression involving the Riemann zeta function.
  • In[7]:= Limit[Zeta[1+x, v] - 1/x, x->0]

    Out[7]=

  • The enhanced Limit can find the limit of an expression involving a PolyGamma function.
  • In[8]:= Limit[x^2 PolyGamma[2,x], x->Infinity]

    Out[8]=