THE LIMIT AND CONTINUITY OF A FUNCTION Electronic version of lecture THE LIMIT AND CONTINUITY OF A FUNCTION ELECTRONIC VERSION OF LECTURE Dr Lê Xuân Đại HoChiMinh City University of Technology Faculty[.]
THE LIMIT AND CONTINUITY OF A FUNCTION ELECTRONIC VERSION OF LECTURE Dr Lê Xuân Đại HoChiMinh City University of Technology Faculty of Applied Science, Department of Applied Mathematics Email: ytkadai@hcmut.edu.vn HCM — 2016 Dr Lê Xuân Đại (HCMUT-OISP) THE LIMIT AND CONTINUITY OF A FUNCTION HCM — 2016 / 68 OUTLINE THE LIMIT OF A FUNCTION CONTINUITY LIMITS INVOLVING INFINITY ASYMPTOTES DISCONTINUITY MATL AB Dr Lê Xuân Đại (HCMUT-OISP) THE LIMIT AND CONTINUITY OF A FUNCTION HCM — 2016 / 68 The limit of a function Model real-world situations PHYSICS According to the special theory of relativity developed by Albert Einstein, the length of a moving object, as measured by an observer at rest, shrinks as its speed increases If L0 is the length of the object when it is at rest, then its length L, as measured by an observer at rest, when traveling s at speed v(m/s) is given by the formula L = L0 v2 − , where c is the c speed of light Question: If the space shuttle were able to approach the speed of light, what would happen to its length L? Dr Lê Xuân Đại (HCMUT-OISP) THE LIMIT AND CONTINUITY OF A FUNCTION HCM — 2016 / 68 The limit of a function Dr Lê Xuân Đại (HCMUT-OISP) Model real-world situations THE LIMIT AND CONTINUITY OF A FUNCTION HCM — 2016 / 68 The limit of a function Model real-world situations We need to find s lim L0 v→c−0 v − = L0 c s c2 1− = c Conclusion: The closer the speed of the shuttle is to the speed of light, the closer the length of the shuttle, as seen by an observer at rest, gets to Dr Lê Xuân Đại (HCMUT-OISP) THE LIMIT AND CONTINUITY OF A FUNCTION HCM — 2016 / 68 The limit of a function Definition of the limit of a function Let f (x) be defined on some open interval that contains the number a, except possibly at a itself DEFINITION 1.1 The number L ∈ R is called the limit of f (x) as x approaches a, and we write lim f (x) = L x→a if for every number ε > there is a number δ > such that if < |x − a| < δ then |f (x) − L| < ε lim f (x) = L means that the values of f (x) can be made x→a as close as we please to L by taking x close enough to a (but not equal to a) Dr Lê Xuân Đại (HCMUT-OISP) THE LIMIT AND CONTINUITY OF A FUNCTION HCM — 2016 / 68 The limit of a function Dr Lê Xuân Đại (HCMUT-OISP) Definition of the limit of a function THE LIMIT AND CONTINUITY OF A FUNCTION HCM — 2016 / 68 The limit of a function Dr Lê Xuân Đại (HCMUT-OISP) Definition of the limit of a function THE LIMIT AND CONTINUITY OF A FUNCTION HCM — 2016 / 68 The limit of a function Calculating Limits using the limit laws CALCULATING LIMITS USING THE LIMIT LAWS THEOREM 1.1 Suppose that lim f (x) = A ∈ R and lim g(x) = B ∈ R Then x→a x→a lim[f (x) ± g(x)] = A ± B x→a lim[c.f (x)] = c.A, where c is a constant x→a lim[f (x).g(x)] = A.B x→a f (x) A = if B 6= x→a g(x) B lim Dr Lê Xuân Đại (HCMUT-OISP) THE LIMIT AND CONTINUITY OF A FUNCTION HCM — 2016 / 68 The limit of a function Calculating Limits using the limit laws EXAMPLE 1.1 x3 − 5x + Evaluate lim x→3 x2 − SOLUTION (x3 − 5x + 4) x3 − 5x + lim x→3 lim = = x→3 x2 − lim(x2 − 2) x→3 33 − 5.3 + 16 = = 32 − Dr Lê Xuân Đại (HCMUT-OISP) THE LIMIT AND CONTINUITY OF A FUNCTION HCM — 2016 10 / 68