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I have a question regarding problem 3a on page 9 in Omalley's Singular Perturbation Methods for ODEs, regarding what he called "Friedrich's problem".

I am not sure how did they get the asymptotic relation: $x(t,\epsilon) \sim (\exp(1-t)-\exp(1-t/\epsilon))$ as $\epsilon \to 0$ uniformly in $t \in [0,1]$.

I get that the solution is

\begin{gather}\nonumber x(t,\epsilon)=(1/(\exp((-1+\sqrt{1-4\epsilon})/(2\epsilon))-\exp((-1-\sqrt{1-4\epsilon})/(2\epsilon)))\\ \nonumber (\exp(((-1+\sqrt{1-4\epsilon})/(2\epsilon))t)-\exp(((-1-\sqrt{1-4\epsilon})/(2\epsilon))t)).

\end{gather}

Now if I am not mistaken I need to show that: $\lim_{\epsilon \to 0, \\ t \in [0,1]} x(t,\epsilon)/(e^{1-t}-e^{1-t/\epsilon})=1$

The part of exponents with $-\sqrt{1-4\epsilon}$ vanishes when $\epsilon \to 0$, and $e^{-t/\epsilon}\to 0$ as $\epsilon \to 0$, but other than that I don't see how to show that the limit approaches 1.

Obviously there's l'HOPITAL there, but I don't see how many times should I use l'HOPITAL?

For those who don't have the book I'll iterate the problem:

3a.

Consider the two-point problem:

$$\epsilon \ddot{x}+\dot{x}+x=0 , t\in [0,1], $$

$$x(0)=0, x(1)=1$$

Determine the exact solution and show that:

$x(t,\epsilon)\sim e^{1-t}-e^{1-t/\epsilon}$ as $\epsilon \to 0$ uniformly in $t\in [0,1]$.

Any pointers on how to compute the limit?