Infinite limits and limits at infinity
See why 1x2 grows huge near zero, draw a horizontal asymptote, and open up the forms ∞ ÷ ∞ and ∞ − ∞.
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Guide
If f(x) grows without bound as x comes close to a number, we write limf(x) = ∞. There is no limit as a number, but the notation describes what happens; the graph has a vertical asymptote there.
When x itself gets larger and larger, the limit can be a number: lim(x→∞)f(x) = L. Then the line y = L is a horizontal asymptote.
∞ ÷ ∞ and ∞ − ∞ are not numbers; they only say there is still work to do. For a fraction of polynomials divide by the highest power of x; for a difference of roots multiply by the conjugate.