Error evaluating limx→0x−tan⁡xx3 I solved it like this, limx→0(1x2−tan⁡xx3)=limx→0(1x2−tan⁡xx⋅1x2) Now using the property limx→0tan⁡xx=1 we have: limx→0(1x2−1x2)=0 Please explain my error! How can I avoid such errors?

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Error evaluating limx→0x−tan⁡xx3  I solved it like this,  limx→0(1x2−tan⁡xx3)=limx→0(1x2−tan⁡xx⋅1x2)  Now using the property  limx→0tan⁡xx=1  we have:  limx→0(1x2−1x2)=0  Please explain my error! How can I avoid such errors?

stamptsk · Accepted Answer

The explanation you&#039;re looking for is this. You are implicitly using the properties lim(f+g)=lim(f)+lim(g) and lim(fg)=lim(f)lim(g), but this is only true when all the limits in these equalities exist, (disclaimer: there are important assumptions that I&#039;m not writing but that should accompany these properties). More specifically, what you did (implicitly) was: limx→0(1x2−tan⁡xx3)=limx→0(1x2−tan⁡xx⋅1x2) =limx→0(1x2)+limx→0(−tan⁡xx⋅1x2) (Incorrect) =limx→0(1x2)−limx→0(tan⁡xx)limx→0(1x2) (Incorrect)  =limx→0(1x2)−limx→0(1x2)(∗) =limx→0(1x2−1x2) (Incorrect) =0 (**) (*) As correct as something meaningless can be (**) Actually correct, but it&#039;s too late

Damian Roberts · Accepted Answer

Although, in fact, limx→0tan⁡xx=1 you cannot deduce from that that  limx→01x2−tan⁡xx×1x2=limx→01x2−1x2  In this case, LHopitals Rule is the way to go:  limx→0x−tan⁡xx3=limx→0−tan2x3x2    =−13limx→0tan⁡xx2    =−13

Error evaluating \lim_{x\to 0}\frac{x-\tan x}{x^3} I solved it like this, \lim_{x\to 0}

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