I understand that this is a very stupid question but I'm not getting the answer.
At $x=\pi/2$, what is the value of $tan(x)$? Should it be $-\infty$ or $+\infty$?
Text tells it to be $+\infty$. But why?
Geometrically thinking, it comes out to be $+\infty$. But how to explain the graph which has both the values at $\pi/2$?
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$\begingroup$It shouldn't be anything because it's not defined there.
$\endgroup$ $\begingroup$Instead of trying to "evaluate" $\tan(\pi/2)$ by simply observing its graph, why not go back to good old-fashioned trig arguments? If you were to evaluate the value of $\tan(\pi/2)$ on the unit circle, you would essentially be constructing this argument: $$ \tan(\pi/2) = \frac{\sin(\pi/2)}{\cos(\pi/2)} = \frac{1}{0} $$ Note that is the same as asking for the $y$-coordinate of the point (0,1) on the unit circle to be divided by its $x$-coordinate. Due to various reasons (see ; or Numberphile's wonderfully informative video Problems with Zero ), mathematicians have not defined what it means to divide by zero because the value is not consistent in various contexts. So division by zero is simply "undefined" - which leads to your uncertainty as to whether $\tan(\pi/2) = +\infty$ or if $\tan(\pi/2)=-\infty$ (!)
$\endgroup$ $\begingroup$There are two common ways to "compactify" the reals.
The first is the one you're probably more familiar with. You add two additional points, $+\infty$ and $-\infty$, located at the two "ends" of the number line. This is called the extended real line. Commonly, the notation $\infty$ is used instead of $+\infty$: thast causes some confusion with....
The second is called the projective real line. It adds only single point $\infty$, which is located on both ends of the line (think of the two ends being glued together to form a circle). This is closely related to the Riemann sphere and "complex infinity". This construction tends to be more useful when you're working with polynomials and rational functions, or when doing complex analysis. If we view $\tan$ as function from the usual real line to the projective real line, then we can continuously extend it so that $\tan(\pi/2) = \infty$.
$\endgroup$ $\begingroup$It depends on how you interpret the situation.
Case 1:-
We see $\displaystyle\tan\frac{\pi}{2}=\frac{\sin\frac{\pi}{2}}{\cos\frac{\pi}{2}}=\frac{1}{0}$ which is undefined in mathematics.
Case 2:-
As you can see from the graph the $\displaystyle\lim_{x\rightarrow\frac{\pi}{2}^{\mp}}\tan(x)=+\infty$.
Both cases are equally correct and valid.
Easily speaking it is geometrically infinite and algebraically undefined.
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