#Limits in calculus bc plus#

Let f be a function defined on an open interval containing c with the exception of possibly at c itself and let L be a real number. f(x) oscillates between two fixed values as x approaches c.Įxample: 'Weird function that doesn't make sense when you zoom in and just looks like crazy patterns of zigzags' f(x) increases or decreases without bound as x approaches c.Įxample: There is a vertical asymptote at cģ. f(x) approaches different numbers from the left and right sides of c.Įxample: Piecewise functions that break off at x = cĢ. Analytic Approach - use algebra or calculusĪ limit doesn't always exist and there are ways for you to spot the nonexistence of a limit.Ĭommon Types of Behavior Associated With Nonexistent Limits:ġ. Numerical Approach - construct a table of valuesģ. (Except when we are dealing with one-sided limits, in which x approaches c only from one side.)ġ. Theorem 1.9 - Two Special Trigonometric Limits:Īs x approaches c, the limit L is the number f(x) approaches from both the left and right sides. Then lim (-^- ( x -> c)) f(x) exists and is equal to L. If h(x) ≤ f(x) ≤ g(x) for all x in an open interval containing c, except possibly at c itself, and if Rewrite the fraction by multiplying both the numerator and the denominator by the same radical minus (or plus) the same number not under a radical.

#Limits in calculus bc plus#

When you have a rational function with something under a radical plus (or minus) another number that is not under a radical for its numerator or denominator, you can rewrite the fraction to enhance the limit solving process. Indeterminate Form - 0/0 - you cannot determinate the limit of the function from this form alone! So rewrite the fraction to avoid the indeterminate form! This will also reduce encounters with the indeterminate form. This will eliminate the values where there is a hole in the graph to make finding the limit easier. When dealing with a rational function, factor both the numerator and the denominator completely and simplify. Reinforce your conclusion with a graph or with a table Apply Theorem 1.7 to conclude analytically thatĤ. (Make sure you use a g where the limit of g(x) CAN be evaluated using direct substitution!)ģ. If the limit can't be evaluated with direct substitution, find a function g that agrees with f for all x other than x = c Recognize which limits can be evaluated by direct substitution (Theorems 1.1 - 1.6)Ģ. If the limit of g(x) as x approaches c exists, then the limit of f(x) also exists andġ.

Let c be a real number and let f(x) = g(x) for all x ≠ c in an open interval containing c. Theorem 1.7 - Functions That Agree at All but One Point:

Let c be a real number in the domain of the given trig function Theorem 1.6 - Limits of Trigonometric Functions: Theorem 1.5 - The Limit of a Composite Function: The following limit is valid for all c if n is odd, and it is valid for c > 0 if n is even: Theorem 1.4 - The Limit of a Function Involving a Radical: If r is a rational function given by r(x) = p(x) / q(x) and c is a real number such that q(c) ≠ 0, then If p is a polynomial function and c is a real number, then Theorem 1.3 - Limits of Polynomial and Rational Functions: Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits: Let b and c be real numbers and let n be a positive integer Direct Substitution can be used to evaluate a limit at c - if the function is continuous at c