What happens to $y=\frac{1}{x}$ as $x$ goes to negative infinity?

Key takeaway: This is a small but important end-behavior question. For reciprocal functions, the sign of $x$ matters, but the limit idea is the same: as the input grows in magnitude, the output shrinks toward zero.

For

$y=\frac{1}{x},$

as $x\to -\infty$, the denominator becomes a very large negative number.

So the fraction gets closer and closer to $0$ from the negative side:

• when $x=-10$, $y=-0.1$
• when $x=-100$, $y=-0.01$
• when $x=-1000$, $y=-0.001$

Therefore,

$y\to 0.$

If you want the signed behavior more precisely, you can say $y\to 0^-$, but many mark schemes accept simply $0$.

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Pitfalls the pros know 👇 A common mistake is to write $y\to\infty$ just because $x\to-\infty$. For reciprocal functions, bigger $|x|$ means smaller $|1/x|$. Another slip is forgetting the sign: here the values stay negative, even though they get very close to zero.

What if the problem changes? If the question were $y=\frac{1}{x}$ as $x\to\infty$, the result would still be $y\to 0$, but now from the positive side. If it were $y=-\frac{1}{x}$ as $x\to-\infty$, then the values would approach $0$ from the positive side instead.

Tags: end behavior, reciprocal function, limit at infinity