The stated result can be easily obtained by successive applications of integration by parts. We know that 
$$
1 - \Phi(x) = \Phi^c(x) = \frac{1}{\sqrt{2 \pi}} \int_{x}^{\infty} e^{-t^2/2}dt.
$$ To apply integration by parts, multiply and divide the integrand by 
$\frac{1}{t}$ to obtain 
$$
\Phi^c(x) = \frac{1}{\sqrt{2 \pi}} \int_{x}^{\infty} \frac{1}{t} \left(t e^{-t^2/2}\right)dt
$$ 
The terms shown in brackets can be integrated to obtain $-e^{-t^2/2}$. Hence, the integral simplifies to 
$$
\Phi^c(x) = \frac{1}{\sqrt{2\pi}} \frac{e^{-x^2/2}}{x} - \frac{1}{\sqrt{2\pi}} \int_{x}^{\infty} \frac{e^{-t^2/2}}{t^2}.
$$
Now we know that 
* $\phi(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2}$ and moreover 
* the integral term on the right hand side is always greater than 0, i.e. specifically, 
$$
\frac{1}{\sqrt{2\pi}} \int_{x}^{\infty} \frac{e^{-t^2/2}}{t^2} \geq 0.
$$ 

Hence, combining these observations we find that 
$$
\Phi^c(x) \leq \frac{\phi(x)}{x}
$$ or in other words, 
$$
\frac{1 - \Phi(x)}{\phi(x)} \leq \frac{1}{x}
$$

Now, you could continue to apply integration by parts again on $\Phi^c(x)$ to obtain a lower bound. More specifically, we obtain 
$$
\Phi^c(x) = \phi(x) \left( \frac{1}{x} - \frac{1}{x^3} \right) + \int_{x}^{\infty} 3t^{-4} e^{-t^2/2}dt.
$$ Again, the integral on the right hand side is positive and hence, 
$$
\phi(x) \left( \frac{1}{x} - \frac{1}{x^3} \right) \leq \Phi^c(x).
$$
Combining both these inequalities we find that $\frac{1}{x} - o(x^{-1}) \leq \frac{\Phi^c(x)}{\phi(x)} \leq \frac{1}{x}$. In other words, 
$$
\frac{\Phi^c(x)}{\phi(x)} \approx \frac{1}{x}
$$ in the asymptotic limit of $x$.

It turns out that more applications of integration by parts leads to an alternating series which is enveloping. More details can be taken from the textbook by Christopher G Small, *Expansions and asymptotics for statistics*, Monographs on Statistics and Applied Probability 115. Boca Raton, FL: CRC Press, ISBN 978-1-58488-590-0/hbk; 978-1-4200-1102-9/ebook, pp. xiv+343 (2010), [MR2681183](https://mathscinet.ams.org/mathscinet-getitem?mr=2681183), [Zbl 1196.62002](https://zbmath.org/1196.62002), which should be available online.