Question

how to evaluate lim x approach 0 ..(x-tan^-1)/x^3??

Answer 1

@Miaaziz (x- arc tan (x))/x^3?

Answer 2

yes

Answer 3

ok. here comes the solution

Answer 4

@Miaaziz

Answer 5

thank you . but how do you get (x-x^3/3 + x^5/5.....)/x^3 ?

Answer 6

Thats the Gregory series expansion of arc tan x

Answer 7

arc tan x= x-x^3/3 +x^5/5-x^7/7+....

Answer 8

oh i see. thanks . you help me a lot

Answer 9

\[\lim_{x \rightarrow 0}\frac{ x-arc \tan x }{ x^3 },\frac{ 0 }{ 0 }~form\]
\[=\lim_{x \rightarrow 0}\frac{ 1-\frac{ 1 }{ 1+x^2 } }{ 3x^2 }\]
\[=\lim_{x \rightarrow 0}\frac{ 1+x^2-1 }{ \left( 1+x^2 \right)3x^2 }\]
\[=\lim_{x \rightarrow 0}\frac{ x^2 }{ \left( 1+x^2 \right) 3x^2}\]
\[=\lim_{x \rightarrow 0}\frac{ 1 }{ 3\left( 1+x^2 \right) }=\frac{ 1 }{ 3 }\]