AP Calculus BC Unit 1 Limits and Continuity

University:
Ohio State University
Course:
MATH 1151 | Calculus I
Academic year:

2021
Views:

104

Pages:

3
Author:

saireerowboz

AP CALCULUS BC SIMPLE STUDIES REVIEW Unit 1: Limits and Continuity ● Limit: the value that a function (typically denoted as ‘f(x)’) approaches as it gets closer and closer to a certain ‘x’ value ○ Ask yourself: “As ‘x’ approaches a value, what is f(x) getting closer to? ○ Typically, to find the limit of a function, you can just plug in ‘x’ to f(x) ■ Ex. What is the 𝑙𝑖𝑚 of 𝑥 2 ? 𝑥→2 ● All you do is plug x (which = 2) into f(x) (which is 𝑥 2 ) ○ 22 = 4 = 𝑙𝑖𝑚−𝑥 2 𝑥→2 ● Left hand limit: this is the value of f(x) as it approaches the value of x from the LEFT side ○ Denoted as: 𝑙𝑖𝑚− 𝑥→2 ■ The ‘-’ superscript indicates a LEFT HAND LIMIT ■ To find the answer, determine what the value that f(x) approaches as x = 1.9, 1.99, 1.999, etc. ● Notice how the x values are less than 2, but are getting closer and closer to the actual value of 2. ● Right hand limit: this is the value of f(x) as it approaches a value of x from the RIGHT side. ○ Denoted as 𝑙𝑖𝑚+ 𝑥→2 ■ The ‘+’ superscript indicates a RIGHT HAND LIMIT. ■ To find the answer, determine what the value that f(x) approaches as x = 2.1, 2.01, 2.001, etc. ● notice how the x values are greater than two, but are getting closer and closer to the actual value of 2 ● In order for a limit to exist, the LH and RH limits have to be equal to each other. If not, the limit does not exist (indicated as DNE). ○ Ex. 𝑙𝑖𝑚1/x 𝑥→0 ■ The LH limit = ‘-∞’ but the RH limit = ‘+∞’. Thus lim x→ 0 of 1/x = DNE ○ *REMEMBER: If the limit of a function equals ‘+∞’, that doesn’t mean that it doesn’t exist. As long as both the RH and LH limits equal ‘+∞’, then the limit can and does exist. ○ If given a graph, you can determine if the limit exists if there aren’t any breaks in the graphs. ● If a question asks for the limit of f(x) as x→ ‘∞’, this is just asking “what value does f(x) approach as x gets bigger and bigger. ○ Ex. lim x→ ∞ of 1/x ■ As you plug in greater values of x (x value gets larger and larger), the function gets closer and closer to 0, because 1/(a bigger number) gets smaller (closer to 0). Thus, the answer would be 0. ● When a function is a polynomial divided by another polynomial and x→ ∞ : ○ If the highest power of x for the numerator is greater than the denominator, then the limit of that function as x→ ∞ is ∞ ○ If the highest power of x for both the numerator and denominator are equal, then the limit of that function as x→ ∞ is the coefficient of the highest power of x in the numerator divided by the coefficient of the highest power of x in the denominator. ○ If the highest power of x for the numerator is less than the denominator, then the limit of that function as x→ ∞ is 0. ● Continuity: A function is continuous ONLY if the LH limit = RH limit = f(x) ○ The limit of a function has to exist, and it has to equal the value of the function at that ‘x’ value. ■ Ex. lim x→ 2of 𝑥 2 = 4 (the limit exists because the RH limit and the LH limit are equal) and f(2) = 4. The limit x→ 2 and f(2) are both equal to 4, so the function is CONTINUOUS. ○ If given a graph, you can determine if it’s continuous if there are no breaks in the graph, or open circles on that function on the graph. ■ Basically, the function on the graph looks continuous, with no holes in it.