I can understand why $${\lim_{θ \to 90} \tan(θ) = ∞}$$
But since $$\tan90°=\frac{\sin90°}{\cos90°}$$
which results in $\frac{1}{0}$, I get confused, as $\frac{1}{0}$ could be both positive and negative infinity, which is one of the ways I think of its undefined nature. In such case, why do we consider the positive infinity scenario?
Thanks in advance. :)
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$\begingroup$$\frac{1}{0}$ is undefined, and same is the case with $\tan 90^{\circ}$.
To show why, use the definition of division. It says $\frac{a}{b}=c$ when $a=bc$ holds and $a$ is unique when $b,c$ are kept fixed. But note that in case of $0$, you have $a \cdot 0= c$ holds whenever you set $c=0$ and $a\in \mathbb{N}$, a contradiction to uniqueness.
$\endgroup$ 1 $\begingroup$If $\theta$ is exactly $90^\circ$, then $\tan \theta$ is undefined because of division by zero.
If $\theta$ tends to $90^\circ$, namely $\theta -90^\circ$ is (non-zero) infinitesimal, then $\tan\theta=\infty$.
You said:
as $\frac{1}{0}$ could be both positive and negative infinity
This is correct only when you clearly understand its meaning. $\frac{1}{0}$ is not valid as an arithmetic expression, it is only valid as a symbol, denoting that the numerator tends to $1$ and the denominator is infinitesimal. Just as when we speak of a "$\frac{0}{0}$ indeterminate", we actually mean the limit status of an expression of the form $\frac{\text{infinitesimal}}{\text{infinitesimal}}$.
$\endgroup$ $\begingroup$I put an answer here a couple of years ago that may help: Finding the Possible Values of y = tan(x).
The gist: as $\theta \rightarrow \dfrac {\pi}{2}$, the values grow to $\infty$ except for $\dfrac {\pi}{2}$.
$\tan \dfrac {\pi}{2}$ is undefined due to the definition of tangent, i.e. $\dfrac {\sin \frac {\pi}{2}}{\cos \frac {\pi}{2}} = \dfrac {1}{0}$ (and similarly for $\dfrac {3\pi}{2}$, except $\sin \dfrac {3 \pi}{2}$ is $-1$).
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