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Limits at ∞ x → ∞ means that x gets larger and larger without a bound. Oktay Olmez and Serhan Varma Calculus Lecture 2 1/1 Limits at ∞ x → ∞ means that x gets larger and larger without a bound. x → −∞ means that x gets smaller and smaller without a bound. Oktay Olmez and Serhan Varma Calculus Lecture 2 1/1 Limits at ∞ x → ∞ means that x gets larger and larger without a bound. x → −∞ means that x gets smaller and smaller without a bound. Oktay Olmez and Serhan Varma Calculus Lecture 2 1/1 Limits at ∞ x → ∞ means that x gets larger and larger without a bound. x → −∞ means that x gets smaller and smaller without a bound. lim x→∞ 1 1 = 0 = lim . Note that for a positive number k, x→−∞ x x lim x→∞ Oktay Olmez and Serhan Varma 1 1 = 0 = lim k . x→−∞ x xk Calculus Lecture 2 1/1 Mathematical Definition For f (x) a real function, the limit of f as x approaches infinity is L, denoted lim f (x) = L, x→∞ means that for all ε > 0, there exists c such that |f (x) − L| < ε whenever x > c. Similarly, the limit of f as x approaches negative infinity is L, denoted lim f (x) = L, x→−∞ means that for all ε > 0 there exists c such that |f (x) − L| < ε whenever x < c. For example lim e x = 0. x→−∞ Oktay Olmez and Serhan Varma Calculus Lecture 2 2/1 Oktay Olmez and Serhan Varma Calculus Lecture 2 3/1 An application: Finding the Area of a Circle Oktay Olmez and Serhan Varma Calculus Lecture 2 4/1 An application: Finding the Area of a Circle Oktay Olmez and Serhan Varma Calculus Lecture 2 4/1 An application: Finding the Area of a Circle Let f (n) be the area of the n-gon inscribed in a circle of radius r . Oktay Olmez and Serhan Varma Calculus Lecture 2 4/1 An application: Finding the Area of a Circle Let f (n) be the area of the n-gon inscribed in a circle of radius r . Then, Area of a circle with radius r is lim f (n) = πr 2 n→∞ Oktay Olmez and Serhan Varma Calculus Lecture 2 4/1 Math Dance Moves Oktay Olmez and Serhan Varma Calculus Lecture 2 5/1 Math Dance Moves Oktay Olmez and Serhan Varma Calculus Lecture 2 5/1 Infinite Limits Oktay Olmez and Serhan Varma Calculus Lecture 2 6/1 Infinite Limits Oktay Olmez and Serhan Varma Calculus Lecture 2 6/1 Infinite Limits lim f (x) = ∞ means that f (x) can be made as large as we wish by x→c + taking x sufficiently close but to the right of c. Oktay Olmez and Serhan Varma Calculus Lecture 2 7/1 Infinite Limits lim f (x) = ∞ means that f (x) can be made as large as we wish by x→c + taking x sufficiently close but to the right of c. lim+ f (x) = −∞ means that f (x) can be made as small as we wish by x→c taking x sufficiently close but to the right of c. Oktay Olmez and Serhan Varma Calculus Lecture 2 7/1 Infinite Limits lim f (x) = ∞ means that f (x) can be made as large as we wish by x→c + taking x sufficiently close but to the right of c. lim+ f (x) = −∞ means that f (x) can be made as small as we wish by x→c taking x sufficiently close but to the right of c. lim f (x) = ∞ means that f (x) can be made as large as we wish by x→c − taking x sufficiently close but to the left of c. Oktay Olmez and Serhan Varma Calculus Lecture 2 7/1 Infinite Limits lim f (x) = ∞ means that f (x) can be made as large as we wish by x→c + taking x sufficiently close but to the right of c. lim+ f (x) = −∞ means that f (x) can be made as small as we wish by x→c taking x sufficiently close but to the right of c. lim f (x) = ∞ means that f (x) can be made as large as we wish by x→c − taking x sufficiently close but to the left of c. lim f (x) = −∞ means that f (x) can be made as small as we wish by x→c − taking x sufficiently close but to the left of c. Oktay Olmez and Serhan Varma Calculus Lecture 2 7/1 Oktay Olmez and Serhan Varma Calculus Lecture 2 8/1 Vertical and Horizontal Asymptotes Definition Line x = c is a vertical asymptote of the graph of y = f (x). If any of the following is satisfied: lim f (x) = ∞ x→c + lim f (x) = −∞ x→c + lim f (x) = ∞ x→c − lim f (x) = −∞ x→c − Oktay Olmez and Serhan Varma Calculus Lecture 2 9/1 Vertical and Horizontal Asymptotes Definition Line x = c is a vertical asymptote of the graph of y = f (x). If any of the following is satisfied: lim f (x) = ∞ x→c + lim f (x) = −∞ x→c + lim f (x) = ∞ x→c − lim f (x) = −∞ x→c − Definition Line y = b is a horizontal asymptote of the graph of y = f (x). If any of the following is satisfied: lim f (x) = b x→∞ lim f (x) = b x→−∞ Oktay Olmez and Serhan Varma Calculus Lecture 2 9/1
