Limit laws, continuity conditions, and common limit evaluation techniques for an intro calculus course. Front: the term or rule. Back: a plain-language definition or the rule statement.
35 cards · basic cards · AI-written, checked twice. Edit anything.
- Limit of a function (informal definition)
- The value that f(x) gets arbitrarily close to as x approaches a, regardless of whether f(a) is defined.
- Epsilon-delta definition of a limit
- For all epsilon > 0, there exists delta > 0 such that if 0 < |x - a| < delta, then |f(x) - L| < epsilon.
- Left-hand limit (notation and meaning)
- lim(x->a-) f(x) is the value f(x) approaches as x approaches a from the left (from values less than a).
- Right-hand limit (notation and meaning)
- lim(x->a+) f(x) is the value f(x) approaches as x approaches a from the right (from values greater than a).
- When does a two-sided limit exist?
- A limit lim(x->a) f(x) exists if and only if the left-hand and right-hand limits both exist and are equal.
- Sum law for limits
- If lim(x->a) f(x) = L and lim(x->a) g(x) = M, then lim(x->a) [f(x) + g(x)] = L + M.
- Difference law for limits
- If lim(x->a) f(x) = L and lim(x->a) g(x) = M, then lim(x->a) [f(x) - g(x)] = L - M.
- Product law for limits
- If lim(x->a) f(x) = L and lim(x->a) g(x) = M, then lim(x->a) [f(x) * g(x)] = L * M.
- Constant multiple law for limits
- If lim(x->a) f(x) = L and c is a constant, then lim(x->a) [c * f(x)] = c * L.
- Quotient law for limits
- If lim(x->a) f(x) = L and lim(x->a) g(x) = M with M not equal to 0, then lim(x->a) [f(x)/g(x)] = L/M.
- Power law for limits
- If lim(x->a) f(x) = L and n is a positive integer, then lim(x->a) [f(x)]^n = L^n.
- Root law for limits
- If lim(x->a) f(x) = L and n is a positive integer (with L >= 0 for even n), then lim(x->a) (f(x))^(1/n) = L^(1/n).
- Infinite limit definition
- lim(x->a) f(x) = infinity means for all M > 0, there exists delta > 0 such that if 0 < |x - a| < delta, then f(x) > M.
- Vertical asymptote
- A vertical line x = a is a vertical asymptote of f if lim(x->a+) f(x) = infinity, lim(x->a-) f(x) = infinity, or one equals negative infinity.
- Horizontal asymptote
- A horizontal line y = L is a horizontal asymptote of f if lim(x->infinity) f(x) = L or lim(x->-infinity) f(x) = L.