حد و صورتهای مبهم
Limits & Indeterminate Forms
محاسبهٔ حد با بسط تیلور و قاعدهٔ هوپیتال.
ریاضی عمومی کنکور ارشد علوم کامپیوتر
کارتها · 10
A function continuous on R with finite limits at both +inf and -inf is uniformly continuous.
If f is continuous and one-to-one on an interval, it must be strictly increasing or strictly decreasing — and then it maps the interval onto another interval of the same type.
To find a root of a continuous function, take the supremum of the set where it is still negative — continuity squeezes the value there to exactly zero from both sides.
When a limit has a variable-limit integral ∫_0^x g(t) dt on top of a 0/0 (or ∞/∞ ) quotient, differentiate it with the FTC — its derivative is just g(x) — and the integral disappears.
Replace each piece of a 0/0 quotient by its Maclaurin series, keep terms up to the first surviving power, and read the limit off the leading coefficients.
A monotone function can only have jump discontinuities, and the set of those jumps is always at most countable — never an interval, never uncountable.
A function into a discrete codomain like Z is continuous exactly where it is locally constant, so its continuity set is open and its discontinuity set is closed.
Two continuous functions that agree on a dense set (like Q ) are equal everywhere; one continuous function with a constant value on a dense set is that constant.
A sum whose number of terms grows with n is squeezed by replacing every term with the largest and smallest term; if both bounds (each n copies of a single term) share a limit, the sum has that limit too.
The set of points at distance zero from A is exactly A , so it is always closed, contains A , and equals A iff A is closed.