= Solution
For a regular <space curve> parametrized by <arc length> $s$, its <curvature> is $\kappa=\|dT/ds\|$, where $T=dx/ds$ is the unit tangent. Differentiating the given parametrization gives
$$
x'(t)=\frac{e^t}{2}(\cos t-\sin t,\ \sin t+\cos t,\ \sqrt2),\qquad \|x'(t)\|=e^t.
$$
The curve approaches the origin as $t\to-\infty$; the origin is a limiting endpoint. Its <arc length> measured from that endpoint is $s=\int_{-\infty}^t e^u\,du=e^t$, so
$$
\boxed{x(s)=\left(\frac s2\cos\log s,\ \frac s2\sin\log s,\ \frac s{\sqrt2}\right),\qquad s>0.}
$$
This is a <logarithmic conical helix>. Its unit tangent and derivative are
$$
T(s)=\frac12(\cos\log s-\sin\log s,\ \sin\log s+\cos\log s,\ \sqrt2),\qquad T'(s)=\frac1{2s}(-\sin\log s-\cos\log s,\ \cos\log s-\sin\log s,\ 0).
$$
Consequently,
$$
\boxed{\kappa(s)=\frac1{\sqrt2\,s}.}
$$
The infinitely many turns near the limiting origin have finite total <arc length>, while the <curvature> diverges there.
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