Is $\frac{x^2-3x-4}{-3x-15}\Big/\frac{x^2-16}{x^2-x-30}$ defined at $x=6$ or not?

Question

If a number is in the denominator of a rational expression that itself is the denominator of another rational expression, such that the number causes the latter denominator to be undefined, then isn't the entire expression undefined? For example, isn't $$\frac{\frac{x^2-3x-4}{-3x-15}}{\frac{x^2-16}{x^2-x-30}}=\frac{\frac{(x-4)(x+1)}{-3(x+5)}}{\frac{(x+4)(x-4)}{(x-6)(x+5)}}$$ undefined at $x=6\;?$ $$\frac{\frac{x^2-3x-4}{-3x-15}}{\frac{x^2-16}{x^2-x-30}} \implies\quad\frac{(x-4)(x+1)(x-6)(x+5)}{-3(x+5)(x+4)(x-4)},x\neq{-5},{-4},4,6\quad?$$ If so, why do my scientific calculator and Desmos compute the expression as zero at $x=6\;?$

Answer 1

$$\frac{\quad\frac{x^2-3x-4}{-3x-15}\quad}{\frac{x^2-16}{x^2-x-30}} \implies\quad\frac{(x-4)(x+1)(x-6)(x+5)}{-3(x+5)(x+4)(x-4)},x\neq{-5},{-4},4,6\quad?$$

Rather: $$f(x)=\frac{\quad\frac{x^2-3x-4}{-3x-15}\quad}{\frac{x^2-16}{x^2-x-30}}\iff\bigg(\color\red{x\neq-5,4,6}\quad\text{and}\quad f(x)=\frac{(x+1)(x-6)}{-3(x+4)} \bigg).$$

isn't $$\frac{\quad\frac{x^2-3x-4}{-3x-15}\quad}{\frac{x^2-16}{x^2-x-30}}$$ undefined at $x=6\;?$

This expression is indeed undefined at $x=6;$ Wolfram also agrees.

why do my scientific calculator and Desmos compute the expression as zero at $x=6\;?$

Most likely, Desmos determines a graphing function's domain only after getting rid of any compound (multi-storey) fraction: $$\frac{\quad\frac{x^2-3x-4}{-3x-15}\quad}{\frac{x^2-16}{x^2-x-30}}$$ contains four implicit conditions $(\color\red{x\ne{-}5,{-}4,4,6}),$ while $$\frac{(x-4)(x+1)(x-6)(x+5)}{-3(x+5)(x+4)(x-4)}$$ contains only three implicit conditions $(\color\red{x\ne{-}5,{-}4,4}).$

Because of this, Desmos is selectively noticing the compound fraction's implicit conditions, mistaking $f(6)$ as $0$ even while it doesn't similarly think that $f(-5)= \frac{44}3$ or that $f(4)=\frac5{12}.$

Answer 2

Whether or not an 'expression' is 'undefined' is not a straightforward rigorous question. That's basically what you are finding out. e.g. How about $$\frac{x^2}{x}?$$ Is this 'undefined' at $x=0$? Or to put it in line with your example, I do an even dumber one, e.g. How about $$\frac{1}{\tfrac{1}{x}}\ ?$$ 'Expressions' are ambiguous in the sense that they are not the same as 'functions'. The latter is the more grown up notion once you start being clear about what function you are considering and what it's domain is things become clearer (and you can talk about e.g. things like 'removable singularities' etc. etc.)

I really want to say do you mean the function $f : \mathbf{R} \to \mathbf{R}$ given by $x \mapsto x$ or do you mean $g : \mathbf{R}\setminus \{0\} \to \mathbf{R}$ given by $x \mapsto 1/(1/x)$. Of course now you see that $f = g$ on $\mathbf{R}\setminus \{0\}$ and that $g$ can be extended to a continuous function on the whole of $\mathbf{R}$ but putting it like this just dispels anything interesting or ambiguous.

The calculators/graphing software computes it like that probably because it does something like simplifying it algebraically before evaluating it.