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

Jett Castaneda

Jett Castaneda

Answered question


Error evaluating limx0xtanxx3
I solved it like this,
Now using the property
we have:
Please explain my error! How can I avoid such errors?

Answer & Explanation



Beginner2022-01-27Added 23 answers

The explanation you'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'm not writing but that should accompany these properties). More specifically, what you did (implicitly) was:
=limx0(1x2)+limx0(tanxx1x2) (Incorrect)
=limx0(1x2)limx0(tanxx)limx0(1x2) (Incorrect)

=limx0(1x21x2) (Incorrect)
=0 (**)
(*) As correct as something meaningless can be
(**) Actually correct, but it's too late

Damian Roberts

Damian Roberts

Beginner2022-01-28Added 14 answers

Although, in fact, limx0tanxx=1 you cannot deduce from that that
In this case, LHopitals Rule is the way to go:

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