Let be $X$ a random variable that obeys a standard normal distribution and $t>0$. We exclude the set where $X$ attains $0$. Compute $P(\frac{1}{X}\leq t)$ and express it by the standard normal distribution function $\Phi$. Then, calculate $\lim\limits_{t\to0}P(\frac{1}{X}\leq t)$.
$$ \begin{align*} P(\frac{1}{X}\leq t)&=P(X\geq \frac{1}{t},X>0)+P(X\leq \frac{1}{t},X<0)\\&=\int\limits_{\frac{1}{t}}^{\infty}f_X(x)dx+\int\limits_{-\infty}^0f_X(x)dx\\&=1-\Phi\left(\frac{1}{t}\right)+\Phi(0).\end{align*} $$
Now, we immediately see $\lim\limits_{t\to0}P(\frac{1}{X}\leq t)=\lim\limits_{t\to0}(1-\Phi\left(\frac{1}{t}\right)+\Phi(0))=\Phi(0)=\frac{1}{2}.$
Is this correct?