How do you find the limit of limx→0tan⁡x−sin⁡xx3?My attempt:limx→0tan⁡x−sin⁡xx3limx→01x2(tan⁡xx−sin⁡xx)limx→01x2(1−1)limx→01x2×00However, according to wolframalpha and my book, I'm wrong.

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How do you find the limit of limx→0tan⁡x−sin⁡xx3?My attempt:limx→0tan⁡x−sin⁡xx3limx→01x2(tan⁡xx−sin⁡xx)limx→01x2(1−1)limx→01x2×00However, according to wolframalpha and my book, I&#039;m wrong.

twineg4 · Accepted Answer

Your way is wrong because you are taking the limit not at once for the whole expression and this is not allowed in general.   We have that  tan⁡x−sin⁡xx3=tan⁡x−sin⁡xsin3xsin3xx3

veiga34 · Accepted Answer

The error lies in the equality limx→01x2(tan⁡xx−sin⁡xx)=limx→01x2(1−1) You cannot take the limit at 0 in part of your expression and leave the x in the remaining expression. You have limx→0tan⁡x−sin⁡xx3=limx→0(x+13x3+O(x4))−(x−16x3+O(x4))x3 =limx→012x3+O(x4)x3 =12

nick1337 · Accepted Answer

limx→0tan⁡x−sin⁡xx3(00)=limx→0sec2⁡x−cos⁡x3x2(00)=limx→02sec2⁡xtan⁡x+sin⁡x6x=16(2⋅1+1)=12

How do you find the limit of \lim_{x \to 0}

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