Limits at. x means that x gets larger and larger without a bound. Oktay Olmez and Serhan Varma Calculus Lecture 2 1 / 1

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1 Limits at x means that x gets larger and larger without a bound. Oktay Olmez and Serhan Varma Calculus Lecture 2 1 / 1

2 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

3 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

4 Limits at x means that x gets larger and larger without a bound. x means that x gets smaller and smaller without a bound. 1 lim x x = 0 = lim x 1. Note that for a positive number k, x lim x 1 x k = 0 = lim x 1 x k. Oktay Olmez and Serhan Varma Calculus Lecture 2 1 / 1

5 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 x ex = 0. Oktay Olmez and Serhan Varma Calculus Lecture 2 2 / 1

6 Oktay Olmez and Serhan Varma Calculus Lecture 2 3 / 1

7 An application: Finding the Area of a Circle Oktay Olmez and Serhan Varma Calculus Lecture 2 4 / 1

8 An application: Finding the Area of a Circle Oktay Olmez and Serhan Varma Calculus Lecture 2 4 / 1

9 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

10 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, 2 Area of a circle with radius r is lim f (n) = πr n Oktay Olmez and Serhan Varma Calculus Lecture 2 4 / 1

11 Math Dance Moves Oktay Olmez and Serhan Varma Calculus Lecture 2 5 / 1

12 Math Dance Moves Oktay Olmez and Serhan Varma Calculus Lecture 2 5 / 1

13 Infinite Limits Oktay Olmez and Serhan Varma Calculus Lecture 2 6 / 1

14 Infinite Limits Oktay Olmez and Serhan Varma Calculus Lecture 2 6 / 1

15 Infinite Limits lim f (x) = means that f (x) can be made as large as we wish by taking x sufficiently close but to the right of c. Oktay Olmez and Serhan Varma Calculus Lecture 2 7 / 1

16 Infinite Limits lim f (x) = means that f (x) can be made as large as we wish by 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 taking x sufficiently close but to the right of c. Oktay Olmez and Serhan Varma Calculus Lecture 2 7 / 1

17 Infinite Limits lim f (x) = means that f (x) can be made as large as we wish by 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 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

18 Infinite Limits lim f (x) = means that f (x) can be made as large as we wish by 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 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

19 Oktay Olmez and Serhan Varma Calculus Lecture 2 8 / 1

20 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) = lim f (x) = lim f (x) = x c lim f (x) = x c Oktay Olmez and Serhan Varma Calculus Lecture 2 9 / 1

21 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) = lim f (x) = 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 x lim x Oktay Olmez and Serhan Varma Calculus Lecture 2 9 / 1