On "Introduction to Discrete Dynamical Systems and Chaos" by Mario Martelli, at 86 pp, it's stated the implication used as title for this question: $$"\ldots\sqrt{3} - 0m = 0 \text{ , which can be true only if } m=\infty."$$ but I don't understand the algebra of this reasoning. The expression is equivalent to $\sqrt{3} = 0$, that should not be always false on the Real line? Introducing $0m$ and setting $m=\infty$ isn't rising an indeterminate form $0\infty$?
Thank you in advance
The question discussed in the book is about the tangent lines of $$ 0=G(x,y)=x^2-xy+y^2-1 $$ in several points. First $(x_0,y_0)=(1,1)$ is discussed for the regular case, then the special situation in the point $(x_0,y_0)=(2/\sqrt3,1/\sqrt3)$ is explored. As the equation of the tangent line in general is $$ 0=(2x_0-y_0)(x-x_0)+(2y_0-x_0)(y-y_0), $$ one sees that this tangent is vertical because of $2y_0-x_0=0$. That the slope $m=dy/dx$ is $∞$, that is, no finite slope exists, just means that the tangent can not be written as a function of $x$.