Product of two perpendicular slopes are always equal to $-1$? We know that product of two slopes is $-1$, but the slope of the $x$ axis is zero and slope of the $y$ axis is infinity. $x$ & $y$ are perpendicular but their product of slopes are not equal to $-1$. Why?
 A: There are lots of cases where expressions of the form $$\dfrac{0}{0},\ \dfrac{\infty}{\infty},\ 0\times\infty$$ may be in some reasonable sense evaluated using limits. 
As an example of the first case we have $$\lim\limits_{x\rightarrow 0} \dfrac{\sin x}{x} =1.$$
Of the second case there are expressions such as $$\lim\limits_{x\rightarrow \infty} \dfrac{ax^4 + 3x^3 -2x^2 - e}{bx^4 + \pi} = \dfrac{a}{b}.$$
And of the final case we have the product of the gradients of two perpendicular lines as they are rotated to coincide with the $x$ and $y$ coordinate axes
$$\lim\limits_{\theta\rightarrow 0}\tan(\theta)\tan\left(\theta + \dfrac{\pi}{2}\right)=\lim\limits_{\theta\rightarrow 0} -\tan(\theta) \cot(\theta) = -1$$
A: The product of 0 and infinity is undefined, since infinity is not considered to be a real number. 
A: If you draw the straight lines $y = \tan \theta \, x$ and $y = -\frac{1}{\tan \theta} x$ then you should be able to see that the lines are perpendicular to each other. Now let $\theta \to 0$.
A: I know that it is an old question (more than 6 years, 4 months), and it is a good question. But I feel the answers (posted here) are not so good. So I am going to answer this old question as per my knowledge.
Answer: The principles of multiplication are not applicable to any quantity with infinitive magnitude. So the hypothesis for the product of slope of $x$-axis and $y$-axis not applicable as per the law of multiplication.
But in case of two lines being perpendicular have their slopes respectively $\tan\phi$ and $\tan(90+\phi)\implies$ product of their slopes $\tan\phi\cdot\tan(90+\phi)=\tan\phi\cdot(-\cot\phi)=-1$.
It is important to note that,
$\bf{(i)}$ Slope of the Y-axis isn't infinity. Slope of the Y-axis is undefined.
$\bf{(ii)}$ The remark about product of slopes of two perpendicular lines being $-1$ only applies when both slopes are actual real numbers (i.e., finite values). Also, the slope of a vertical line is simply not defined, so there is neither paradox nor any "indeterminate form" here.
