Fri Feb 7 00:08:19 MST 1997 euler% math Mathematica 3.0 for Solaris Copyright 1988-96 Wolfram Research, Inc. -- Terminal graphics initialized -- In[1]:= In[2]:= In[3]:= (* ----------[ M a t h e m a t i c a ]---------- *) 0. Second In[4]:= (* ---------- Initialization ---------- *) 0. Second In[5]:= (* ---------- Limits ---------- *) 0. Second In[6]:= (* Start with a famous example => e *) 0. Second In[7]:= Limit[(1 + 1/n)^n, n -> Infinity] 0.44 Second Out[7]= E In[8]:= (* => 1/2 *) 0. Second In[9]:= Limit[(1 - Cos[x])/x^2, x->0] 0.02 Second 1 Out[9]= - 2 In[10]:= (* See Dominik Gruntz, _On Computing Limits in a Symbolic\ > Manipulation System_, Ph.D. dissertation, Swiss Federal Institute of Technology, Zurich, Switzerland, 1996. => 5 *) 0. Second In[11]:= Limit[(3^x + 5^x)^(1/x), x->Infinity] 7.17 Second x x 1/x Out[11]= Limit[(3 + 5 ) , x -> Infinity] In[12]:= (* => 1 *) 0. Second In[13]:= Limit[Log[x]/(Log[x] + Sin[x]), x->Infinity] 5.09 Second Log[x] Out[13]= Limit[---------------, x -> Infinity] Log[x] + Sin[x] In[14]:= (* => - e^2 [Gruntz] *) 0. Second In[15]:= Limit[(Exp[x*Exp[-x]/(Exp[-x] + Exp[-2*x^2/(x + 1)])] - Exp[x])/x,\ > x->Infinity] 11.21 Second 2 x -x (-2 x )/(1 + x) x x/(E (E + E )) -E + E Out[15]= Limit[--------------------------------------, x -> Infinity] x In[16]:= (* => 1/3 [Gruntz] *) 0. Second In[17]:= Limit[x*Log[x]*Log[x*Exp[x] - x^2]^2/Log[Log[x^2 +\ > 2*Exp[Exp[3*x^3*Log[x]]]]], x->Infinity] 18.5 Second x 2 2 x Log[x] Log[E x - x ] Out[17]= Limit[------------------------, x -> Infinity] 3 3 x x 2 Log[Log[2 E + x ]] In[18]:= (* => 1/e [Knopp, p. 73] *) 0. Second In[19]:= Limit[1/n * n!^(1/n), n->Infinity] 1 3 Series::esss: Essential singularity encountered in Gamma[- + 1 + O[n] ]. n 1 3 Series::esss: Essential singularity encountered in Gamma[- + 1 + O[n] ]. n 1 3 Series::esss: Essential singularity encountered in Gamma[- + 1 + O[n] ]. n General::stop: Further output of Series::esss will be suppressed during this calculation. 0.19 Second 1/n n! Out[19]= Limit[-----, n -> Infinity] n In[20]:= (* Rewrite the above problem slightly => 1/e *) 0. Second In[21]:= Limit[1/n * Gamma[n + 1]^(1/n), n->Infinity] 0.18 Second 1/n Gamma[1 + n] Out[21]= Limit[---------------, n -> Infinity] n In[22]:= (* => 1 [Gradshteyn and Ryzhik 8.328(2)] *) 0. Second In[23]:= Limit[Gamma[z + a]/Gamma[z]*Exp[-a*Log[z]], z->Infinity] 0.03 Second Gamma[a + z] Out[23]= Limit[------------, z -> Infinity] a z Gamma[z] In[24]:= (* => e^z [Gradshteyn and Ryzhik 9.121(8)] *) 0. Second In[25]:= Limit[HypergeometricPFQ[{1, k}, {1}, z/k], k->Infinity] 0.61 Second z Out[25]= E In[26]:= (* => Euler's_constant [Gradshteyn and Ryzhik 9.536] *) 0. Second In[27]:= Limit[Zeta[x] - 1/(x - 1), x->1] 0.07 Second Out[27]= EulerGamma In[28]:= (* => gamma(x) [Knopp, p. 385] *) 0. Second In[29]:= Limit[n^x/(x * Product[(1 + x/k), {k, 1, n}]), n->Infinity] 0.09 Second x n n! Out[29]= Limit[----------------------, n -> Infinity] x Pochhammer[1 + x, n] In[30]:= (* See Angus E. Taylor and W. Robert Mann, _Advanced Calculus_,\ > Second Edition, Xerox College Publishing, 1972, p. 125 => 1 *) 0. Second In[31]:= Limit[x * Integrate[Exp[-t^2], {t, 0, x}]/(1 - Exp[-x^2]), x->0] 0.53 Second Out[31]= 1 In[32]:= (* => [-1, 1] *) 0. Second In[33]:= {Limit[x/Abs[x], x->0, Direction -> 1], Limit[x/Abs[x], x->0,\ > Direction -> -1]} 0.15 Second Out[33]= {-1, 1} In[34]:= (* => pi/2 [Richard Q. Chen] *) 0. Second In[35]:= Limit[ArcTan[-Log[x]], x->0, Direction -> -1] 0.14 Second Pi Out[35]= -- 2 In[36]:= (* Try again after loading Calculus`Limit` *) 0. Second In[37]:= << Calculus`Limit` 0.93 Second In[38]:= (* Start with a famous example => e *) 0. Second In[39]:= Limit[(1 + 1/n)^n, n -> Infinity] 0.18 Second Out[39]= E In[40]:= (* => 1/2 *) 0. Second In[41]:= Limit[(1 - Cos[x])/x^2, x->0] 0.12 Second 1 Out[41]= - 2 In[42]:= (* See Dominik Gruntz, _On Computing Limits in a Symbolic\ > Manipulation System_, Ph.D. dissertation, Swiss Federal Institute of Technology, Zurich, Switzerland, 1995. => 5 *) 0. Second In[43]:= Limit[(3^x + 5^x)^(1/x), x->Infinity] 0.28 Second Out[43]= E In[44]:= (* => 1 *) 0. Second In[45]:= Limit[Log[x]/(Log[x] + Sin[x]), x->Infinity] 0.22 Second Out[45]= 1 In[46]:= (* => - e^2 [Gruntz] *) 0. Second In[47]:= Limit[(Exp[x*Exp[-x]/(Exp[-x] + Exp[-2*x^2/(x + 1)])] - Exp[x])/x,\ > x->Infinity] 0.93 Second Out[47]= 0 In[48]:= (* => 1/3 [Gruntz] *) 0. Second In[49]:= Limit[x*Log[x]*Log[x*Exp[x] - x^2]^2/Log[Log[x^2 +\ > 2*Exp[Exp[3*x^3*Log[x]]]]], x->Infinity] 1 Power::infy: Infinite expression - encountered. 0 1 Power::infy: Infinite expression - encountered. 0 1 Power::infy: Infinite expression - encountered. 0 General::stop: Further output of Power::infy will be suppressed during this calculation. Infinity::indet: Indeterminate expression ComplexInfinity + <<5>> + ComplexInfinity encountered. Infinity::indet: Indeterminate expression ComplexInfinity + <<26>> + ComplexInfinity encountered. 42.86 Second Out[49]= Indeterminate In[50]:= (* => 1/e [Knopp, p. 73] *) 0. Second In[51]:= Limit[1/n * n!^(1/n), n->Infinity] 0.24 Second 1 Out[51]= - E In[52]:= (* Rewrite the above problem slightly => 1/e *) 0. Second In[53]:= Limit[1/n * Gamma[n + 1]^(1/n), n->Infinity] 0.02 Second Out[53]= 0 In[54]:= (* => 1 [Gradshteyn and Ryzhik 8.328(2)] *) 0. Second In[55]:= Limit[Gamma[z + a]/Gamma[z]*Exp[-a*Log[z]], z->Infinity] 0.49 Second Out[55]= 1 In[56]:= (* => e^z [Gradshteyn and Ryzhik 9.121(8)] *) 0. Second In[57]:= Limit[HypergeometricPFQ[{1, k}, {1}, z/k], k->Infinity] 0.24 Second z Out[57]= E In[58]:= (* => Euler's_constant [Gradshteyn and Ryzhik 9.536] *) 0. Second In[59]:= Limit[Zeta[x] - 1/(x - 1), x->1] 0.43 Second Out[59]= EulerGamma + Log[2] + Log[Pi] - Log[2 Pi] In[60]:= Simplify[%] 0.01 Second Out[60]= EulerGamma In[61]:= (* => gamma(x) [Knopp, p. 385] *) 0. Second In[62]:= Limit[n^x/(x * Product[(1 + x/k), {k, 1, n}]), n->Infinity] 0.62 Second Gamma[1 + x] Out[62]= ------------ x In[63]:= FullSimplify[%] 0.24 Second Out[63]= Gamma[x] In[64]:= (* See Angus E. Taylor and W. Robert Mann, _Advanced Calculus_,\ > Second Edition, Xerox College Publishing, 1972, p. 125 => 1 *) 0. Second In[65]:= Limit[x * Integrate[Exp[-t^2], {t, 0, x}]/(1 - Exp[-x^2]), x->0] 0.61 Second Out[65]= 0 In[66]:= (* => [-1, 1] *) 0. Second In[67]:= {Limit[x/Abs[x], x->0, Direction -> 1], Limit[x/Abs[x], x->0,\ > Direction -> -1]} 0.2 Second Out[67]= {-1, 1} In[68]:= (* => pi/2 [Richard Q. Chen] *) 0. Second In[69]:= Limit[ArcTan[-Log[x]], x->0, Direction -> -1] 0.16 Second Pi Out[69]= -- 2 In[70]:= (* ---------- Quit ---------- *) 0. Second In[71]:= Quit[] real 96.77 user 93.78 sys 0.70