Infinite limits, limits at infinity and asymptotes
Distinguish unbounded outputs from faraway inputs, and describe the lines a graph approaches.
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.
