Continuity, infinite limits and asymptotes

Continuity and Limits

f is continuous at a if and only if lim(x→a) f(x) = f(a): the limit must exist, be finite, and coincide with the value of the function. This is the definition that directly links the two concepts.

Extension by continuity

If f is not defined at a but lim(x→a) f(x) = L exists and is finite, we can define a new function equal to f everywhere else, and equal to L at a. Example: f(x) = sin(x)/x is not defined at 0, but we extend it by setting f(0) = 1.

Discontinuities

• First type: the left-hand and right-hand limits exist and are finite, but differ from one another or from f(a). Example: the integer part E(x) for any integer x.
• Second type: at least one of the lateral limits does not exist or is infinite. Example: sin(1/x) at 0.

Asymptotes defined by limits

• Horizontal asymptote: lim(x → ±∞) f(x) = L (line y = L)
• Vertical asymptote: lim(x → a) f(x) = ±∞ (line x = a)
• Oblique asymptote: lim(x → +∞) [f(x) – (mx + p)] = 0, where m = lim f(x)/x and p = lim (f(x) – mx)

Common pitfall

The Intermediate Value Theorem assumes continuity on a CLOSED and bounded interval: overlooking this leads to false ‘proofs’ of the existence of a solution. Similarly, an oblique asymptote exists only if BOTH limits (m and then p) are finite; if m is infinite or does not exist, there is no oblique asymptote.