Some simple example that illustrates that the condition 
$$\lim_{|x|\to \infty} f(x) = 0 \quad (1)$$ 
is not necessary. If the condition were necessary $f\in L^2$ would imply that the limit in (1) holds. 

Take in dimension 1 the function
$$
f(x) = \sum_{n=1}^{\infty} \chi_{I_n}(x)
$$
where $\chi_{I_n}$ is the characteristic function of the interval $I_n = [-\frac{1}{n^2}, \frac{1}{n^2}]$ then the integral evaluates to
$$
\int |f(x)|^2 dx = \sum_{n=1}^{\infty} |I_n| = \sum_{n=1}^{\infty} \frac{2}{n^2} < \infty\ .
$$
But the function does not converge to zero for $|x|\to \infty$.