Is there a way to evaluate this limit:limx→0sin⁡(etan2⁡x−1)cos35⁡(x)−cos⁡(x)

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davelucasnp0j · Accepted Answer

You should know the following limits:limy→0sin⁡yy=1limy→0ey−1y=1limy→0tan⁡yy=1limy→0(1+y)θ−1y=θ(∀θ∈R)limy→01−cos⁡yy2=12which can be proved using only elementary Calculus tools (i.e. without any Differential Calculus technique). These five limits are usually written as asymptotic relations in the following manner:sin⁡y≈y (1)ey−1≈y (2)tan⁡y≈y (3)(1+y)θ−1≈θy (4)1−cos⁡y≈12y2 (5)as y→0. Using asymptotics (1) - (5) you find:sin⁡(etan2x−1)≈etan2x−1 by (1)≈tan2x by (2)≈x2 by (3)cos35(x)−cos⁡(x)=((1+(cos⁡x−1))35−1)+(1−cos⁡x)≈35(cos⁡x−1)+(1−cos⁡x) by (4)=25(1−cos⁡x)≈15x2 by (5)hence:limx→0sin⁡(etan2{x}−1)cos35(x)−cos⁡(x)=limx→0x215 x2=5;.

Is there a way to evaluate this limit:

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