Limiting x->∞: (e^(x^-2)-1)/(π-2arctan(x^2))

[Tanishq Nandan]

Homework Statement

Find limit for x tending to infinity of: (e^(x^-2)-1)/(2arctan(x^2) - pi)

Homework Equations

arctanx +arccotx=pi/2
arctanx=arccot(1/x)

The Attempt at a Solution

P.S I haven't written limit before each step, and hope you can get how I arrived at the first step.I just took 2 common from the denominator and used simple inverse trig.

However,the answer is -1/2
Where am I going wrong?

 

[member 587159]

$\frac{1}{\cot^{-1}(x^2)} = \tan^{-1}(x^2)$ is not true.

I didn't check whether you made other mistakes though.

 

[Mark44]

Homework Statement

Find limit for x tending to infinity of: (e^(x^-2)-1)/(2arctan(x^2) - pi)

Homework Equations

arctanx +arccotx=pi/2
arctanx=arccot(1/x)

The Attempt at a Solution

View attachment 205654
P.S I haven't written limit before each step, and hope you can get how I arrived at the first step.I just took 2 common from the denominator and used simple inverse trig.

However,the answer is -1/2
Where am I going wrong?

Do you know L'Hopital's Rule?

 

[Tanishq Nandan]

$\frac{1}{\cot^{-1}(x^2)} = \tan^{-1}(x^2)$ is not true.

I didn't check whether you made other mistakes though.

Ok

 

[Tanishq Nandan]

Yes,I do

Do you know L'Hopital's Rule?

 

[Tanishq Nandan]

Ok,I got where I am going wrong,but how to go right?
It looks like we won't be able to use inverse trig

 

[member 587159]

Ok,I got where I am going wrong,but how to go right?
It looks like we won't be able to use inverse trig

Use l'Hopital's rule on the first step you wrote down and you should be fine.

 

[SammyS]

Ok,I got where I am going wrong,but how to go right?
It looks like we won't be able to use inverse trig

Can you find the derivative of $\ \tan^{-1}(x^2) - \frac \pi 2 \,,\$ and derivative of $\ e^{1/x^2}-1\,?$

 

[Tanishq Nandan]

Yeah,I can..and I did,thanks

 

[member 587159]

Yeah,I can..and I did,thanks

Did you get the right answer?

 

[Tanishq Nandan]

Yep,-1/2 (I used another inverse property in b/w though,I had gone wrong in writing the last step,that way I didn't need l'hospital)