Unit 9 Introduction to the Limit Process and a Preview of Calculus

Unit 9 Introduction to the Limit Process and a Preview of Calculus ● Calculate the slope of the secant line using average rate of change from point (1,2) to (3,10) ○ Use the slope formula: 𝑚 = ○ 𝑚= 10−2 3−1 𝑦2 −𝑦1 𝑥2 −𝑥1 8 =2=4 ● You can use the slope of secant line to approximate the slope of the tangent line Finding Limits Graphically and Numerically ● Limits are paths that the graph is approaching. ● Example: consider 𝑓(𝑥) = 𝑥 3 −1 𝑥−1 ○ If you substitute x = 1 into the function its domain would be undefined x 0.75 0.9 0.99 0.999 1 1.001 1.01 1.1 1.25 f(x) 2.3125 2.71 2.9701 2.997001 3? 3.003001 3.0301 3.31 3.8125 ● Limit Notation: ○ 𝑙𝑖𝑚𝑓(𝑥) = 𝐿 𝑥→𝑐 ○ As x approaches “c”, f(x) approaches “L” ○ The limit of f(x) is L as x approaches point ( or hole) from both directions ● Formal Definition of a Limit: Let f be a function defined on an open interval containing c (except possibly at c) and let L be a real number. The statement: ○ 𝑙𝑖𝑚𝑓(𝑥) = 𝐿 𝑥→𝑐 ○ Means that for each 𝜀 > 0 there exists a 𝛿 > 0 such that if ■ 0 < |𝑥 − 𝑐| < 𝛿 then |𝑓(𝑥) − 𝐿| < 𝜀 ■ 𝜀 = epsilon ■ 𝛿= delta Evaluating Limits Analytically ● 𝑙𝑖𝑚𝑏 = 𝑏 𝑥→𝑐 ● 𝑙𝑖𝑚𝑥 = 𝑐 𝑥→𝑐 ● 𝑙𝑖𝑚𝑥 𝑛 = 𝑐 𝑛 𝑥→𝑐 ● Theorem: let b and c be real numbers, let n be a positive integer and let f and g be functions with following limits. ○ 𝑙𝑖𝑚𝑓(𝑥) = 𝐿 𝑥→𝑐 and 𝑙𝑖𝑚𝑔(𝑥) = 𝐾 𝑥→𝑐 ● Scalar Multiplication: 𝑙𝑖𝑚[𝑏𝑓(𝑥)] = 𝑏𝐿 𝑥→𝑐 ● Sum or Difference: 𝑙𝑖𝑚[𝑓(𝑥) + 𝑔(𝑥)] = 𝐿 ± 𝐾 𝑥→𝑐 ● Product: 𝑙𝑖𝑚[𝑓(𝑥)𝑔(𝑥)] = 𝐿𝐾 𝑥→𝑐 𝑓(𝑥) 𝐿 ● Quotient: 𝑙𝑖𝑚 𝑔(𝑥) = 𝐾 𝐾 ≠ 0 𝑥→𝑐 ● Power: 𝑙𝑖𝑚[𝑓(𝑥)]𝑛 = 𝐿𝑛 𝑥→𝑐 ● Example: 𝑙𝑖𝑚(4𝑥 2 + 3) = 𝑥→2 ○ 𝑙𝑖𝑚4𝑥 2 + 𝑙𝑖𝑚3 𝑥→2 𝑥→2 2 ○ 4(𝑙𝑖𝑚4) + 𝑙𝑖𝑚3 𝑥→2 𝑥→2 ○ 4(2)2 + 3 ○ 16 + 3 ○ Answer: 19 ● Direct Substitution Theorem: If p is a polynomial function and c is real number, then ○ 𝑙𝑖𝑚𝑝(𝑥) = 𝑝(𝑐) 𝑥→𝑐 ● Example: 𝑙𝑖𝑚𝑡𝑎𝑛𝑥 𝑥→0 ○ 𝑡𝑎𝑛(0) = 0 Continuity and One-Sided Limits ● A function f is continuous at c if the following 3 conditions are met: ○ f(c) is defined ○ 𝑙𝑖𝑚𝑓(𝑥)exists 𝑥→𝑐 ○ 𝑙𝑖𝑚𝑓(𝑥) = 𝑓(𝑐) 𝑥→𝑐 1 ● Example: 𝑓(𝑥) = 𝑥 Continuous (-∞,0) U (0,∞) ● Theorem: Let f be a function and let c and L be real numbers. The limit of f(x) as x approaches c is L if and only if ○ 𝑙𝑖𝑚 𝑓(𝑥) = 𝐿 𝑥→𝑐 − 𝑙𝑖𝑚 𝑓(𝑥) = 𝐿 and 𝑥→𝑐 − Left Right ● A function is continuous on the closed interval [a,b] if it is continuous on the open interval (a,b) and ○ 𝑙𝑖𝑚 𝑓(𝑥) = 𝑓(𝑎) 𝑥→𝑎− and 𝑙𝑖𝑚 𝑓(𝑥) = 𝑓(𝑏) 𝑥→𝑏 − ● Theorem: If b is a real number and f and g are continuous at x = c, then the following functions are also continuous at c 1. Scalar Multiple: 𝑏𝑓 2. Sum and Difference: 𝑓 ± 𝑔 3. Product: 𝑓𝑔 𝑓 4. Quotient: 𝑔 ● Theorem: If g is continuous at c and f is continuous at g(x=c), then the composite function given by 𝑓 ∘ 𝑔)(𝑥) = 𝑓(𝑔(𝑥))is continuous at c ○ 𝑙𝑖𝑚𝑓(𝑔(𝑥)) 𝑥→𝑐 ○ 𝑓(𝑥)is continuous at 𝑔(𝑐) ○ 𝑔(𝑥)is continuous at 𝑥 = 𝑐 ● Intermediate Value Theorem: If 𝑓is continuous on the closed interval [a,b] and k is any value between 𝑓(𝑎)and 𝑓(𝑏), then there is at least one number c in [a,b] such that f(c) = k ○ 𝑓(𝑎) < 𝑘 < 𝑓(𝑏) ○ 𝑓(𝑎) > 𝑘 > 𝑓(𝑏) Infinite Limits ● Infinite Limits: Let f be a function that is defined at every real number in some open interval containing c (except possibly at c). An “infinite limit” means that either ○ 𝑙𝑖𝑚𝑓(𝑥) = ∞ 𝑥→𝑐 of 𝑙𝑖𝑚𝑓(𝑥) = −∞ 𝑥→𝑐 ● Vertical Asymptote: The line x = c is a vertical asymptote of the graph of f if f(x) approaches infinity (or negative infinity) as x approaches c from the right or the left ○ ○ 𝑙𝑖𝑚 𝑓(𝑥) = ±∞ 𝑥→𝑐− 𝑙𝑖𝑚 𝑓(𝑥) = ±∞ 𝑥→𝑐 + ● Properties of Infinite Limits ○ Let 𝑏 an 𝐿 be real numbers and let 𝑓 and𝑔 be function with the following limits ■ 𝑙𝑖𝑚𝑓(𝑥) = ∞ 𝑙𝑖𝑚𝑔(𝑥) = 𝐿 of 𝑥→𝑐 𝑥→𝑐 ○ Sum of difference: 𝑙𝑖𝑚[𝑓(𝑥) + 𝑔(𝑥)] = ∞ 𝑥→𝑐 ○ Product: 𝑙𝑖𝑚[𝑓(𝑥)𝑔(𝑥)] = ∞ L > 0 𝑥→𝑐 ○ Product: 𝑙𝑖𝑚[𝑓(𝑥)𝑔(𝑥)] = −∞ L < 0 𝑥→𝑐 𝑔(𝑥) ○ Quotient: 𝑙𝑖𝑚 𝑓(𝑥) = 0 𝑥→𝑐 𝐿 ∞ Limits at Infinity ● 𝑙𝑖𝑚 𝑓(𝑥) = 𝐿 𝑙𝑖𝑚 𝑓(𝑥) = 𝐿 𝑥→∞ 𝑥→−∞ ● Definition: The line y = L is a horizontal asymptote of the graph f if ○ 𝑙𝑖𝑚 𝑓(𝑥) = 𝐿 𝑥→∞ of 𝑙𝑖𝑚 𝑓(𝑥) = 𝐿 𝑥→−∞