Connecting Limits at Infinity and Horizontal Asymptotes

As x runs off toward infinity, the terms in a function that grow fastest dominate everything else. A term like 5/x or 7/x² shrinks toward 0 once x is huge, so it stops mattering. What survives is the ratio of the leading behaviors, and that ratio is often a constant — which is exactly what a horizontal asymptote is: the height the graph settles toward (not necessarily reaches) as it runs off to the right or left forever.

The mechanical trick is to divide every single term, top and bottom, by the highest power of x appearing in the denominator. This rewrites the function so every lower-order term becomes a fraction with x in the denominator, and those fractions all vanish as x goes to infinity. What's left is a clean limit you can read off directly. When a square root is involved, remember √(x²) equals |x|, not x — so for x approaching negative infinity, √(x² + c) behaves like -x, not x. That sign flip is what lets a function have two different horizontal asymptotes, one for each direction.

This topic is the infinite partner to limits at a point: same idea of approaching a value without necessarily arriving, just played out as x itself grows without bound instead of sliding toward a finite number. It sets up end-behavior analysis for rational, radical, and later exponential functions, and it previews why horizontal asymptotes describe long-run trends rather than hard boundaries the graph must respect.