Why does the google calculator give $\tan 90^{\circ} = 1.6331779e^{+16}$? I typed in $\tan 90^{\circ}$ in Google and it gave $1.6331779\mathrm{E}16$. How did it come to this answer? Limits? Some magic?
The closest IEEE-754 double value to $\pi/2$ is $1.5707963267948965579989817342720925807952880859375$. The cosine of that, on standard x86_64 hardware evaluates to $6.123233995736766 \times 10^{-17}$. The reciprocal of that is $1.633123935319537 \times 10^{16}$.
Because it seems that Google Calculator works internally


in radians instead of degrees
with floating-point arithmetic instead of symbolic computation


The use of radians means that GC is using an angle of $\frac{\pi}{2}$ radians.  The nearest representable double value is $1.5707963267948965579989817342720925807952880859375$.  I shall denote this approximated value by $\frac{\pi}{2} - \epsilon$, where $\epsilon \approx 6.123233995736766 \times 10^{-17} $ (calculated by using a higher-precision value of $\pi$).

Recall that $\tan (\frac{\pi}{2} - \epsilon) = \cot \epsilon = \frac{1}{\tan \epsilon} $.  On my machine, this evaluates to $1.633123935319537 \times 10^{16}$.  (The  small-angle approximation of $\cot \epsilon \approx \frac{1}{\epsilon}$ happens to give the same answer.)  This is close to what Google Calculator returned, but differs by 33 ppm.