Limit of 1/x as x Approaches Infinity

How the value follows

Nothing changes on top while the bottom grows past every bound, so the quotient has to shrink past every positive bound. At $x = 100$ the value is $0.01$, at $x = 10^{6}$ it is $10^{-6}$, and there is no stage at which it settles anywhere but 0.

The generalisation carries the same argument. Any positive power works, and the larger the power the faster the collapse.

The other end behaves the same way in value: $\lim_{x \to -\infty} \frac{1}{x} = 0$, with the values negative and rising toward 0 from below. So $y = 0$ is a horizontal asymptote in both directions.

Why one over infinity is not an evaluation

You cannot substitute $\infty$, because $\infty$ is not a number in the real system. Writing $\frac{1}{\infty} = 0$ is shorthand for the end-behaviour argument, not a step of arithmetic, and treating it as arithmetic is what produces claims like $\frac{\infty}{\infty} = 1$.

The honest version of the statement is about eventual size: for any tolerance you pick, past some point the values of $\frac{1}{x}$ stay inside it. Pick $0.001$ and every $x > 1000$ qualifies. Pick $10^{-9}$ and every $x > 10^{9}$ qualifies.

$\lim_{x \to 0} \frac{1}{x}$ is a different question with a different answer, and the function is the same. The approach point is doing all the work.

Where this fact does the work

Limits of rational functions at infinity all reduce to this one. Divide the numerator and denominator by the highest power of $x$ in the denominator, and every leftover term that still has an $x$ underneath dies by the fact above.

That is where the degree rule comes from. Equal degrees leave the ratio of leading coefficients, a larger denominator degree leaves 0, and a larger numerator degree leaves something unbounded.

The mistake students make

• Treating $\frac{1}{\infty} = 0$ as arithmetic and extending it, for instance to $\infty - \infty = 0$. That subtraction is an indeterminate form and needs real work, while $\frac{1}{\infty}$ is not, which is why this limit has a settled answer and that one does not.
• Confusing the approach point. $\frac{1}{x} \to 0$ as $x \to \infty$, but $\frac{1}{x}$ blows up as $x \to 0^+$.
• Saying the function reaches 0. The equation $\frac{1}{x} = 0$ has no solution, so the graph never touches its asymptote. A limit describes what the values approach, not what they attain.
• Concluding there is no horizontal asymptote because the limit came out to 0. A limit of 0 is a horizontal asymptote, at $y = 0$.