You also need to distinguish between the use of the entity $\frac{1}{\infty }$ in loose form by applied mathematicians (like me) and proper limits like this $\mathop {\lim }\limits_{x \to \infty } \frac{1}{{{x^n}}};x > 1,n \ge 0$ $\mathop {\lim }\limits_{n \to \infty } \frac{1}{{{x^n}}};x > 1,n \ge 0$ where if you pick the debarred values of x or n you will converge to 1 or 1/0
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