Why Does Tangent Go To Infinity?

[adelin]

Why does tangent go to infinity when it increases 0 to 90 degrees?

 

[SteamKing]

What's the slope of a vertical line?

 

[lurflurf]

because
tan(x)=sin(x)/cos(x)
and
cos(x)->0
sin(x)->1

 

[HallsofIvy]

$$\frac{.999999}{.000001}= 999,999$$
$$\frac{.9999999}{.0000001}= 9,999,999$$
$$\frac{.99999999}{.00000001}= 99,999,999$$
etc.

 

[Millennial]

Well now, tangent doesn't go to infinity at $x=\pi/2$. It depends on which side you approach that value from. If you approach it from the left left, yes, it does go to infinity. But, approaching from the right, you will immediately see that now it goes in the exact opposite direction and becomes negative infinity. In analysis, this is called a singularity.

The tangent function demonstrates a special case of a singularity called a pole, which is the situation when a function behaves like $(z-c)^{-1}$ at some point $z=c$. To see why it is called a pole, you can check a graph of the tangent function in the complex plane.

What you said would be true in the Riemann sphere, which is a specific interpretation of the complex plane as the surface of a sphere. In the Riemann sphere, there is only one infinity and it is neither positive nor negative, just as there is only one zero and zero is neither positive nor negative; so you would not face a problem like this. Complex graphs of these functions demonstrate the poles with only one infinity, unlike real graphs where the function jumps from positive infinity to negative infinity.

 

[HallsofIvy]

Good point.