Just empirically, I believe the OP's $e^{-x}$ example has the property that ray reflections become increasingly vertical, and so will not reach arbitrarily large $x$:

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          ![exconvex][1]

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(I did not, however, perform the calculations to prove this.)

Just empirically, I believe the OP's $e^{-x}$ example has the property that ray reflections quickly become increasingly vertical, and so will not reach arbitrarily large $x$:

---

          ![exconvex][1]

---

(I did not, however, perform the calculations to prove this.)

Just empirically, I believe the OP's example has the property that ray reflections become increasingly vertical, and so will not reach arbitrarily large $x$:

---

          ![exconvex][1]

---

(I did not, however, perform the calculations to prove this.)

Just empirically, I believe the OP's $e^{-x}$ example has the property that ray reflections become increasingly vertical, and so will not reach arbitrarily large $x$:

---

          ![exconvex][1]

---

(I did not, however, perform the calculations to prove this.)