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

  • Thread starter Thread starter Tanishq Nandan
  • Start date Start date
Click For Summary

Homework Help Overview

The discussion revolves around finding the limit as x approaches infinity for the expression (e^(x^-2)-1)/(2arctan(x^2) - pi). The problem involves concepts from calculus, particularly limits and inverse trigonometric functions.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the application of L'Hôpital's Rule and question the validity of certain mathematical identities related to inverse trigonometric functions. There is also an exploration of the steps leading to the limit calculation and where errors may have occurred.

Discussion Status

The discussion is active, with participants providing guidance on using L'Hôpital's Rule and questioning assumptions made in earlier steps. Some participants express uncertainty about the correctness of their approaches and seek clarification on derivatives involved in the limit calculation.

Contextual Notes

There are mentions of specific mathematical identities and rules, such as the relationship between arctan and arccot, which are under scrutiny. Participants also note the potential complications in using inverse trigonometric functions for this limit problem.

Tanishq Nandan
Messages
122
Reaction score
5

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


20170618_003226-1.jpg

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?
 
Physics news on Phys.org
##\frac{1}{\cot^{-1}(x^2)} = \tan^{-1}(x^2)## is not true.

I didn't check whether you made other mistakes though.
 
Tanishq Nandan said:

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?
 
Math_QED said:
##\frac{1}{\cot^{-1}(x^2)} = \tan^{-1}(x^2)## is not true.

I didn't check whether you made other mistakes though.
Ok
 
Yes,I do
Mark44 said:
Do you know L'Hopital's Rule?
 
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
 
Tanishq Nandan said:
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.
 
Tanishq Nandan said:
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\,?##
 
Yeah,I can..and I did,thanks
 
  • #10
Tanishq Nandan said:
Yeah,I can..and I did,thanks

Did you get the right answer?
 
  • #11
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)
 

Similar threads

Puzzling Limit Homework: Solving for x→∞ [π/2-arctan x]^(1/x)
  • · Replies 15 ·
Replies
15
Views
4K
##\lim_{x \to0} \left(\dfrac{1}{\sin^2 x}-\dfrac{1}{x^2}\right)##
  • · Replies 5 ·
Replies
5
Views
2K
Integrate [(x^4+x^2+1)/2(1+x^2)]dx: Soln & Explanation
  • · Replies 2 ·
Replies
2
Views
1K
Solve ##\int\frac{e^{-x}}{x^2}dx## and ##\int \frac{e^{-x}}{x}dx##
  • · Replies 7 ·
Replies
7
Views
2K
Volume integral of x^2 + (y-2)^2 +z^2 = 4 where x , y , z > 0
  • · Replies 21 ·
Replies
21
Views
3K
Show |x^2 sin^8(e^x)| <= (16 pi^3)/3
  • · Replies 7 ·
Replies
7
Views
3K
Limit of x^α.sin²(x)/(x+1) as x approaches infinity is?
  • · Replies 13 ·
Replies
13
Views
3K
Integrate $$y= \frac 1 {\sqrt{2\pi}}e^{ \frac{- x^2} 2},~y=0,~x=0,~x=1$$
  • · Replies 3 ·
Replies
3
Views
2K
Why does dividing by ##\sin^2 x## solve the integral?
  • · Replies 27 ·
Replies
27
Views
4K
How do I change this integral limit from x to t?
  • · Replies 2 ·
Replies
2
Views
2K