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## Homework Statement

Determine the following limit in terms of the two real-valued parameters A and B:

[tex]lim_{x \rightarrow 0} (\frac{Ae^{A/{x^2}}+Be^{B/{x^2}}}{e^{A/{x^2}}+e^{B/{x^2}}})[/tex]

## Homework Equations

L'Hopital's rule

## The Attempt at a Solution

I first divided by [itex]e^{A/{x^2}}[/itex] in both numerator and denominator, and then used L'Hopital's rule to get the limit as B. But this can't be right - surely the limit must be symmetrical in A and B since they are symmetrical in the original expression - and I can easily see how the reverse process would lead me to find the limit as A.

Some working:

[tex]lim_{x \rightarrow 0} (\frac{Ae^{A/{x^2}}+Be^{B/{x^2}}}{e^{A/{x^2}}+e^{B/{x^2}}}) = lim_{x \rightarrow 0} (\frac{A+Be^{B/{x^2}-A/{x^2}}}{1+e^{B/{x^2}-A/{x^2}}}) [/tex]

and A in the numerator and B in the denominator then go to 0 when differentiating using L'Hopital's Rule. Everything else cancels after differentiation to leave B.