- #1
yuiop
- 3,962
- 20
Hi,
I am trying to prove that t goes to zero in the limit that a goes to infinite in this equation:
[tex]1/a \, \, arcsinh(aT)[/tex]
or alternatively that t goes to zero in the limit that v goes to 1 in this equivalent equation:
[tex]t = T \sqrt{(1-v^2)}/v \, \, arcsinh(v/\sqrt{(1 - v^2)})[/tex]
Just for the record, T is physically related to a and v by the following equation:
[tex]T = \frac{v}{a\sqrt{1-v^2}}[/tex]
Now I have tried applying L'Hopital's rule to try and obtain a form that is not indeterminate, but it gets into an alternating pattern that makes it clear that that no matter how many repetitions of the rule are applied it is not going to to resolve to a determinate limit in this form.
Now if I take the hyperbolic sine of both sides I appear to be able to find a limit for sinh(t) which is equal to 0. Is it valid to then claim that the limit of t itself is zero when a=infinite or v=1?
In other words, if it can be shown that:
[tex]\lim_{v\to 1} sinh\left ( t \right ) = 0[/tex]
does it automatically follow that:
[tex]\lim_{v\to 1} t = arsinh( 0) = 0[/tex]
?
If not, any other ideas of how to find the limit of t in this example?
I am trying to prove that t goes to zero in the limit that a goes to infinite in this equation:
[tex]1/a \, \, arcsinh(aT)[/tex]
or alternatively that t goes to zero in the limit that v goes to 1 in this equivalent equation:
[tex]t = T \sqrt{(1-v^2)}/v \, \, arcsinh(v/\sqrt{(1 - v^2)})[/tex]
Just for the record, T is physically related to a and v by the following equation:
[tex]T = \frac{v}{a\sqrt{1-v^2}}[/tex]
Now I have tried applying L'Hopital's rule to try and obtain a form that is not indeterminate, but it gets into an alternating pattern that makes it clear that that no matter how many repetitions of the rule are applied it is not going to to resolve to a determinate limit in this form.
Now if I take the hyperbolic sine of both sides I appear to be able to find a limit for sinh(t) which is equal to 0. Is it valid to then claim that the limit of t itself is zero when a=infinite or v=1?
In other words, if it can be shown that:
[tex]\lim_{v\to 1} sinh\left ( t \right ) = 0[/tex]
does it automatically follow that:
[tex]\lim_{v\to 1} t = arsinh( 0) = 0[/tex]
?
If not, any other ideas of how to find the limit of t in this example?