Figure (1) below illustrates the Jacobi theta function $\vartheta_3\left(0,e^{-\frac{\pi}{x^2}}\right)$ and the linear function $x$ in orange and blue respectively.
Figure (1): Illustration of $\vartheta_3\left(0,e^{-\frac{\pi}{x^2}}\right)$ (orange) and linear function $x$ (blue)
Question: Has it been proven (or can it be) than $\underset{x\to\infty}{\text{lim}}\left(\vartheta_3\left(0,e^{-\frac{\pi}{x^2}}\right)-x\right)=0$?
It is a very good exercice to prove the Fourier series $$f(t)=\sum_n e^{-\pi (t+n)^2 x} = \sum_k x^{-1/2} e^{-\pi k^2/x} e^{2i \pi kt}, \qquad x^{-1/2} e^{-\pi k^2/x} = \int_0^1 f(t) e^{-2i \pi kt}dt$$ from which it follows that $\lim_{x \to \infty}x-\sum_n e^{-\pi n^2 /x^2} = \lim_{x \to \infty} x-x \sum_n e^{-\pi n^2 x^2} = 0$
and that $\lambda(s) = \pi^{-s/2}\Gamma(s/2)\zeta(s)= \lambda(1-s)$