Is there a way to evaluate this limit:

Hallie Watts

Hallie Watts

Answered question

2022-04-14

Is there a way to evaluate this limit:
limx0sin(etan2x1)cos35(x)cos(x)

Answer & Explanation

davelucasnp0j

davelucasnp0j

Beginner2022-04-15Added 16 answers

You should know the following limits:
limy0sinyy=1
limy0ey1y=1
limy0tanyy=1
limy0(1+y)θ1y=θ(θR)
limy01cosyy2=12
which 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:
sinyy (1)
ey1y (2)
tanyy (3)
(1+y)θ1θy (4)
1cosy12y2 (5)
as y0. Using asymptotics (1) - (5) you find:
sin(etan2x1)etan2x1 by (1)
tan2x by (2)
x2 by (3)
cos35(x)cos(x)=((1+(cosx1))351)+(1cosx)
35(cosx1)+(1cosx) by (4)
=25(1cosx)
15x2 by (5)
hence:
limx0sin(etan2{x}1)cos35(x)cos(x)=limx0x215 x2=5;.

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