AP Calculus Unit 1 Limits and Continuity

AP Calculus AB Simple Studies Review Unit 1: Limits and Continuity ● Limit: The value of a function (f(x)) as it gets closer and closer to an X value. o Functional Notation: o Think: As the function approaches an X value, what will the Y value of the function be? ● When do limits fail to exist? o Jumps: occurs when the left and right-hand behavior at a point is different. Jumps typically occur in piecewise functions, where different functions can have different “f(x)” values at jump points. o Unbounded Behavior: occurs when a function approaches infinity or negative infinity (asymptote). o Oscillating Behavior: When a function continues to move between two points ● Limit Properties: used to break apart limit functions and make them easier to solve. Let limx→c = L and limx→c = K ○ Scalar Property of Limits: limx→c [ b * f(x)] = b limx→c [f(x)] = bL ○ Sum and Difference Property of Limits: limx→c [ f(x) ± g(x)] = limx→c f(x) ± limx→c g(x) = L ± K ■ Limits that use plus signs can be represented by the Sum Property ■ Limits that use minus signs can be represented by the Difference Property ○ Product Property of Limits: limx→c [ f(x)*g(x)] = limx→c g(x) * limx→c f(x) = L* K ○ Quotient Property of Limits: limx→c [f(x)/g(x)] = limx→c f(x) / limx→c g(x) = L/K ○ Power Property of Limits: limx→c (f(x))n = (limx→c f(x))n = Ln ● Solving Indeterminate Limits Analytically: If we plug in the C value to f(x), we may get 0/0 which is an indeterminate form, which we cannot have. In order to get a correct answer, we may have to solve the limit function analytically. o Factoring: Factor the numerator and denominator until we have a value that can be cancelled out at the top and bottom. ○ Conjugating: Typically used when there is a square root. ■ By multiplying the numerator and denominator by a conjugate, it will allow you to cancel out values in the top and bottom. ○ Finding a common denominator: This method is typically used when there is a fraction inside of a fraction. ■ By finding a common denominator for the inside fraction, we can then use the “keep, change, flip” method. ○ Synthetic Division: ■ 1) Set up the equation using the root of the binomial and the coefficients of the other function. ■ 2) Bring down the leading coefficient to the bottom row. ■ 3) Multiply that value by the root of the binomial and add it to the next coefficient. ■ 4) Repeat until the last value is added to the coefficient; this is your remainder. o Squeeze Theorem: If h(x) < f(x) < g(x) for the interval, and limx→c h(x) = L and limx→c g(x) = L, then limx→c f(x) = L. o Trig Identities o Special Trig Limits ○ Multiply by a Form of One: In order to cancel some value out, you can multiply by a form of one (# / #). ● Left Hand Limits: the function approaches the x value from the left side. ○ Think that the function approaches a value that is a little bit less than the value. ■ Notation: ■ The “-” indicates that the function is approaching that value from the left ■ Ex: limx→2⁻ f(x) ● In this example approaching two from the left hand side may mean 1.999, 1.9999, and even 1.99999 (getting closer and closer to 2 but never being 2). ● Right Hand Limits: the function approaches the x value from the right side. ○ Think that the function approaches a value that is a little bit more than the value. ■ Notation: ■ The “+” indicates that the function is approaching that value from the right ■ Ex: limx→2⁺ f(x) ● In this example 2+ means 2.01, 2.001, 2.0001 (getting closer and closer to 2 but never being 2). ● For a limit to exist, the Right-hand limit and left-hand limit must be equal. If not, then the limit does not exist (DNE). ● Continuity: Must pass these conditions: ○ 1) F(x) is a function. ○ 2) The limit of f(x) exists. ○ 3) No hole and f(c) = f(x) ○ When a function is continuous, it can be differentiated / it has a derivative throughout the function. ● Discontinuities ○ Removable discontinuity: The limit resulted in a hole but f(c) = f(x). ■ Think: the hole can be “fixed” ● Nonremovable Discontinuity: The function has a break, but the limit does not exist. ■ Happens at an asymptote or jump ● Continuity Discussions o List where discontinuities occur o Show f(c) = limx→c f(x) or that it doesn’t o Explain if the discontinuities are removable or nonremovable and why ● Intermediate Value Theorem: If f(x) is continuous on the closed interval [a,b], and K is between f(a) and f(b) then f(c) = K at least once. ● Vertical Asymptotes: The limit approaches infinity or negative infinity as it approaches the left and right side of an X value. o If solving a limit results in a number over 0, then there is a vertical asymptote at that X point. ○ In order to solve these limits, we must find the behavior of the function to the left and right of the asymptote. ■ First, start with the left side of the asymptote. Think: If I plug in a number a little less than the value of the asymptote, will it be positive or negative? ■ Then do the same with the right side of the asymptote. Think: If I plug in a number a little greater than the asymptote value, will it be positive or negative? ● If both values are positive, then the answer is positive infinity. ● If both values are negative, then the answer is negative infinity. ● If the values are positive and negative, then the answer is DNE. ● Horizontal Asymptotes: describe the end behavior of the left and right ends of a graph, as the limit approaches infinity or negative infinity ○ Horizontal Asymptote Theorem: ■ If the highest degree in the numerator is equal to the highest degree in the denominator, then the limit of that function is equal to the coefficient of the highest degree in the numerator over that in the denominator. ■ If the highest degree of the numerator is bigger than the highest degree in the denominator OR there is a square root in function then, the answer is positive or negative infinity. ● Think: If I plug in the c value (pos. or neg. infinity) to the function, will the answer be positive or negative? ○ If positive, then the answer is positive infinity ○ If negative, then the answer is negative infinity ■ If the highest degree in the numerator is smaller than the highest degree in the denominator, then the limit of that function is equal to zero.