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Here are the definitions and explanations: Definition of Continuity and Limits Continuity* of a function at a point means that the function is defined at that point, its limit exists at that point, and this limit is equal to the function's value at that point. Graphically, a continuous function can be drawn without lifting the pen. The limit* of a function describes the behavior of the function as the input variable approaches a certain value (finite or infinite). Limits of Polynomial Functions For a polynomial function P(x) = a_n x^n + + a_1 x + a_0: At a finite point c: The limit is simply the value of the function at that point. _x c P(x) = P(c) At infinity (± ): The limit is that of its highest degree term. _x ± P(x) = _x ± a_n x^n Limits of Linear Functions A linear function is a special case of a polynomial function of degree 1, f(x) = ax+b: At a finite point c: _x c (ax+b) = ac+b At infinity (± ): _x ± (ax+b) = _x ± ax The limit will be ± depending on the sign of a and the direction of infinity. Limits of an Irrational Function An irrational function contains a variable under a radical (for example, sqrt(x)). At a point c where the function is defined and continuous, the limit is obtained by direct substitution. If direct substitution leads to an indeterminate form (like (0)/(0) or ()/()) or if the point is a boundary of the domain, techniques like multiplying by the conjugate expression can be used to simplify the expression before evaluating the limit. L'Hôpital's Rule L'Hôpital's Rule is used to evaluate limits of indeterminate forms (0)/(0) or ()/(). If _x c f(x) = 0 and _x c g(x) = 0 (or if both tend to ± ), then: _x c (f(x))/(g(x)) = _x c (f'(x))/(g'(x)) provided that the limit of the quotient of the derivatives exists. Limits of Rational Functions A rational function is a quotient of two polynomials, R(x) = (P(x))/(Q(x)): At a finite point c where Q(c) ≠ 0: _x c (P(x))/(Q(x)) = (P(c))/(Q(c)) At a finite point c where Q(c) = 0 and P(c) ≠ 0: The limit is ± , and the sign must be determined by studying the left and right-hand limits. At a finite point c where Q(c) = 0 and P(c) = 0: This is an indeterminate form (0)/(0). One can factor (x-c) from the numerator and denominator, or use L'Hôpital's Rule. At infinity (± ): The limit is that of the ratio of the highest degree terms of the numerator and denominator. _x ± (P(x))/(Q(x)) = _x ± (a_n x^n)/(b_m x^m) That's 2 down. 3 left today — send the next one.