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)))
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
Not really, see you
here
Not really, see you here
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))
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,
Marco
PS: Sorry saw Daniel’s post too late.