Infinite limits, limits at infinity and asymptotes
“The infinite! No other question has ever moved so profoundly the spirit of man.”
Two different questions
An infinite limit describes outputs that grow without bound while the input approaches a point. A limit at infinity describes outputs as the input itself becomes arbitrarily large in magnitude. The output may approach a finite number, grow without bound, or fail to settle.
Infinity describes unbounded behavior; it is not a real-number output. Opposite infinite one-sided limits do not combine into a two-sided infinite limit.
Approaching a vertical asymptote
For \(f(x)=1/x\), a small negative input gives a large negative output; a small positive input gives a large positive output. Squaring the denominator changes that sign behavior.
\(f\)\(x=0\)Initial view
Zoom out vertically to follow the branches as their outputs grow without bound near \(x=0\). The pale band marks the initial view.
Example 1
Check the sign on each side
For \(r(x)=\frac{x+2}{x-1}\), the numerator stays positive near \(1\). The denominator is small and negative on the left, small and positive on the right. Hence
Thus \(x=1\) is a vertical asymptote. The two-sided limit is neither \(+\infty\) nor \(-\infty\), since the directions disagree.
Looking far away
The basic starting point is \(\lim_{x\to\pm\infty}1/x=0\): increasing the magnitude of a denominator makes its reciprocal arbitrarily small. Products then give \(\lim_{x\to\pm\infty}1/x^n=0\) for every positive integer \(n\). The earlier limit laws and squeeze theorem also apply to these approaches, with their usual conditions.
For the next graph, let \(g(x)=\frac{2x^2+x}{x^2+1}\). Dividing by \(x^2\) shows that both tails approach \(2\). The graph even crosses that height at \(x=2\).
\(g\)\(y=2\)Initial view
Move the slider right to zoom out along the input axis. The pale region marks the initial view; the vertical scale stays fixed.
An asymptote describes limiting behavior, not a line that a graph is forbidden to touch.
The same calculation works as \(x\to-\infty\), so \(y=2\) is a horizontal asymptote at both ends. We evaluate the rewritten quotient; infinity is not substituted as a number.
Rule
Rational-function end behavior
For a quotient of nonzero polynomials, compare their degrees (highest powers). As \(x\to\pm\infty\):
Lower numerator degree: the quotient tends to \(0\).
Equal degrees: it tends to the ratio of leading coefficients.
Higher numerator degree: its magnitude grows without bound; there is no horizontal asymptote. Determine the sign at each end.
These conclusions follow by dividing numerator and denominator by the highest denominator power, as in Example 2.
So \(y=x+2\) is a slant asymptote at both ends. Here \(s(x)\) itself tends to \(+\infty\) on the right and \(-\infty\) on the left. A numerator degree exactly one higher than the denominator produces a slant line after division.
\(s\)\(y=x+2\)
The difference \(s(x)-(x+2)=5/(x-2)\) tends to \(0\), even though the function and line are unbounded.
