>>16Isn't -1/infinity a fixed point though?
What do you mean by that?
It's 0.z=x/infinity approaches 0 as x approaches zero
Harder to visualize when it's written like that.
Time to bust out muh
\(\LaTeX\) and BBCode skills
!\(\displaystyle{\lim_{x \to 0} \frac{x}{\infty} = 0}\)But it doesn't even need to be a limit as x approach zero. It could even be a limit at infinity, just as long as the denominator has a higher power in the dominating x term than that of the numerator.
Example:
\(\displaystyle{\lim_{x \to \infty} \frac{x}{x^{2}} = \lim_{x \to \infty} \frac{1}{x} = \frac{1}{\infty} = 0}\)For the above, coefficients don't matter as long as the numerator's dominating term's power is less than that of the denominator.
\(\Large Indeterminate \;\; Forms\)The following are indeterminate forms:
\(\frac{0}{0}\) and
\(\frac{\infty}{\infty}\)However, the above indeterminate forms can be used with
L'Hospital's Rule, at least when dealing with derivatives.
\(\displaystyle{\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{\frac{d}{dx}[f(x)]}{\frac{d}{dx}[g(x)]}} = \)\(\displaystyle{ \lim_{x \to c} \frac{\frac{d}{dx}(\frac{d}{dx}[f(x)])}{\frac{d}{dx}(\frac{d}{dx}[g(x)])}} = \; ... \; = \;\)\(\displaystyle{ \lim_{x \to c} \frac{\frac{d^{n}}{dx^{n}}[f(x)]}{\frac{d^{n}}{dx^{n}}[g(x)]}}\)Not to be confused with the quotient rule, which is different and doesn't apply here.
Other indeterminate forms:
\(0 \times \infty\),
\(1^{\infty}\),
\(\infty - \infty\),
\(0^{0}\), and
\(\infty^{0}\).
But for the other indeterminate forms, you're fucked, unless you can somehow factor shit out of them or find a common denominator to rewrite them as determinate forms (or, just get them into
\(\frac{0}{0}\) or
\(\frac{\infty}{\infty}\)to apply L'Hospital's Rule).
And you can keep on applying that until you get a determinate form (as long as
\(g^{n}(x) \neq 0\)).