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# Why wolfram returns a bad response to a limit?

Posted 10 years ago
 A friend asked me about a limit, use it to check your answer wolfram but ... wolfram returns another value lim (x to -infinity ) ((x+5sqr(4+x²))/((9+8x³)^(1/3)))  Attachments:
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Posted 10 years ago
 Hi,I still think that your calculations incorrect at these places. If $x\rightarrow -\infty$ the nominator is real and the denominator is complex. The whole thing will go complex. Also look a this: Table[Evaluate[(x + 5 Sqrt[4 + x^2])/((9 + 8 x^3)^(1/3))] // N, {x, 0, -100, -1}] {4.8075, 10.1803, 1.59641 - 2.76506 I, 1.27021 - 2.20006 I, 1.15435 - 1.99939 I, 1.0996 - 1.90456 I, 1.06948 - 1.85239 I, 1.05117 - 1.82068 I, 1.03923 - 1.8 I, 1.03102 - 1.78578 I, 1.02514 - 1.77559 I, 1.02078 - 1.76804 I, 1.01746 - 1.7623 I, 1.01488 - 1.75782 I, 1.01283 - 1.75427 I, 1.01117 - 1.75141 I, 1.00982 - 1.74906 I, 1.0087 - 1.74712 I, 1.00776 - 1.74549 I, 1.00696 - 1.74411 I, 1.00628 - 1.74293 I, 1.0057 - 1.74192 I, 1.00519 - 1.74104 I, 1.00475 - 1.74027 I, 1.00436 - 1.7396 I, 1.00402 - 1.73901 I, 1.00371 - 1.73848 I, 1.00344 - 1.73802 I, 1.0032 - 1.7376 I, 1.00298 - 1.73722 I, 1.00279 - 1.73688 I, 1.00261 - 1.73657 I, 1.00245 - 1.7363 I, 1.0023 - 1.73604 I, 1.00217 - 1.73581 I, 1.00205 - 1.7356 I, 1.00194 - 1.7354 I, 1.00183 - 1.73522 I, 1.00174 - 1.73506 I, 1.00165 - 1.73491 I, 1.00157 - 1.73477 I, 1.00149 - 1.73463 I, 1.00142 - 1.73451 I, 1.00136 - 1.7344 I, 1.0013 - 1.73429 I, 1.00124 - 1.7342 I, 1.00118 - 1.7341 I, 1.00113 - 1.73402 I, 1.00109 - 1.73394 I, 1.00104 - 1.73386 I, 1.001 - 1.73379 I, 1.00096 - 1.73372 I, 1.00093 - 1.73366 I, 1.00089 - 1.7336 I, 1.00086 - 1.73354 I, 1.00083 - 1.73349 I, 1.0008 - 1.73343 I, 1.00077 - 1.73339 I, 1.00074 - 1.73334 I, 1.00072 - 1.7333 I, 1.0007 - 1.73326 I, 1.00067 - 1.73322 I, 1.00065 - 1.73318 I, 1.00063 - 1.73314 I, 1.00061 - 1.73311 I, 1.00059 - 1.73308 I, 1.00058 - 1.73305 I, 1.00056 - 1.73302 I, 1.00054 - 1.73299 I, 1.00053 - 1.73296 I, 1.00051 - 1.73294 I, 1.0005 - 1.73291 I, 1.00048 - 1.73289 I, 1.00047 - 1.73286 I, 1.00046 - 1.73284 I, 1.00045 - 1.73282 I, 1.00043 - 1.7328 I, 1.00042 - 1.73278 I, 1.00041 - 1.73276 I, 1.0004 - 1.73275 I, 1.00039 - 1.73273 I, 1.00038 - 1.73271 I, 1.00037 - 1.7327 I, 1.00036 - 1.73268 I, 1.00035 - 1.73267 I, 1.00035 - 1.73265 I, 1.00034 - 1.73264 I, 1.00033 - 1.73262 I, 1.00032 - 1.73261 I, 1.00032 - 1.7326 I, 1.00031 - 1.73259 I, 1.0003 - 1.73257 I, 1.0003 - 1.73256 I, 1.00029 - 1.73255 I, 1.00028 - 1.73254 I, 1.00028 - 1.73253 I, 1.00027 - 1.73252 I, 1.00027 - 1.73251 I, 1.00026 - 1.7325 I, 1.00026 - 1.73249 I, 1.00025 - 1.73248 I} Also - not a proof- you see (Evaluate[(x + 5 Sqrt[4 + x^2])/((9 + 8 x^3)^(1/3)) /. x -> -1000000] // N ) - (1 - I Sqrt[3]) gives 2.50044*10^-12 - 4.33187*10^-12 I Look also at definition of cubic root.Cheers,MarcoPS: Sorry saw Daniel's post too late.
Posted 10 years ago
Posted 10 years ago
 All depends on how that cube root is done. To get surd behavior one requires CubeRoot (or Surd) in Mathematica. In[154]:= Limit[(x + 5*Sqrt[4 + x^2])/CubeRoot[9 + 8*x^3], x -> -Infinity] (* Out[154]= -2 *) In[155]:= Limit[(x + 5*Sqrt[4 + x^2])/(9 + 8*x^3)^(1/3), x -> -Infinity] (* Out[155]= 1 - I Sqrt[3] *) As for Wolfram|Alpha, I do not know offhand what heuristics it uses to determine which was intended. But one can force the surd behavior by using cbrt().lim (x to -infinity ) ((x+5sqr(4+x.b2))/cbrt(9+8x.b3))
Posted 10 years ago
 Not really, see you here
Posted 10 years ago
 Hi,the problem is with your calculation on the board. The solution will be complex because the cubic root in the denominator will go negative as x tends to infinity. The step from your second to the third term and from the third to the fourth appear to be incorrect. Cheers, Marco
Posted 10 years ago
 Not really, see you here