Finding Limit t->0: H'(r) and τ(r)

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SUMMARY

The discussion focuses on evaluating the limit as \( r \) approaches 0 for the function \( \tau(r) = k + \left(\frac{H(r)}{r}\right)^a \), where \( a > 0 \) and \( k > 0 \). Participants highlight the challenge of the 0/0 form when substituting \( r = 0 \) directly into the equation. The role of the slope of \( H(r) \) is emphasized as a critical factor in resolving the limit, with attempts to apply l'Hôpital's rule leading to recursive outcomes.

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urbanist
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Hi all,

If we have
[itex]H'(r)=r+\tau(r)H(r)[/itex]

and

[itex]\tau(r)=k+(H(r)/r)^a[/itex]

where

[itex]a>0, k>0,[/itex] and [itex]H(0)=0[/itex],

can we say anything about [itex]{lim}_{r\rightarrow 0^+}\tau(r)[/itex]?

Thanks a lot!
 
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Sure you can.
Note - the trouble with evaluating the limit just by putting r=0 is the 0/0 in the second term right?
So what role would the slope of H play in reconciling this problem?
 
Yes, that fraction is the problem.

I tried to solve it with l'Hopital's rule, but just got into a recursion, as expected...
 

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