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C o e on ELECTRONIC VERSION OF LECTURE nZ Dr Lê Xuân Đại hV ie HoChiMinh City University of Technology Faculty of Applied Science, Department of Applied Mathematics Email: ytkadai@hcmut.edu.vn in m THE LIMIT AND CONTINUITY OF A FUNCTION Dr Lê Xuân Đại (HCMUT-OISP) HCM — 2016 https://fb.com/sinhvienzonevn THE LIMIT AND CONTINUITY OF A FUNCTION HCM — 2016 / 68 C o e THE LIMIT OF A FUNCTION CONTINUITY LIMITS INVOLVING INFINITY ASYMPTOTES DISCONTINUITY MATL AB hV ie nZ on in m OUTLINE Dr Lê Xuân Đại (HCMUT-OISP) https://fb.com/sinhvienzonevn THE LIMIT AND CONTINUITY OF A FUNCTION HCM — 2016 / 68 C o Model real-world situations PHYSICS ie nZ on e 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 at speed v(m/s) is hV 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? in m The limit of a function Dr Lê Xuân Đại (HCMUT-OISP) https://fb.com/sinhvienzonevn THE LIMIT AND CONTINUITY OF A FUNCTION HCM — 2016 / 68 ie nZ on e C o Model real-world situations hV in m The limit of a function Dr Lê Xuân Đại (HCMUT-OISP) https://fb.com/sinhvienzonevn THE LIMIT AND CONTINUITY OF A FUNCTION HCM — 2016 / 68 e C o Model real-world situations lim L0 v2 − = L0 c nZ v→c−0 on We need to find c2 1− = c hV ie 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 in m The limit of a function Dr Lê Xuân Đại (HCMUT-OISP) https://fb.com/sinhvienzonevn THE LIMIT AND CONTINUITY OF A FUNCTION HCM — 2016 / 68 Definition of the limit of a function C o The limit of a function e Let f (x) be defined on some open interval that contains the number a, except possibly at a itself nZ on 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 hV ie 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 in as close as we please to L by taking x close enough to a) m(but not equal to https://fb.com/sinhvienzonevn Dr Lê Xuân Đại (HCMUT-OISP) THE LIMIT AND CONTINUITY OF A FUNCTION HCM — 2016 / 68 ie nZ on e C o Definition of the limit of a function hV in m The limit of a function Dr Lê Xuân Đại (HCMUT-OISP) https://fb.com/sinhvienzonevn THE LIMIT AND CONTINUITY OF A FUNCTION HCM — 2016 / 68 ie nZ on e C o Definition of the limit of a function hV in m The limit of a function Dr Lê Xuân Đại (HCMUT-OISP) https://fb.com/sinhvienzonevn THE LIMIT AND CONTINUITY OF A FUNCTION HCM — 2016 / 68 C o Calculating Limits using the limit laws e CALCULATING LIMITS USING THE LIMIT LAWS on THEOREM 1.1 Suppose that lim f (x) = A ∈ R and lim g(x) = B ∈ R Then x→a nZ 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 = x→a g(x) B hV ie x→a lim[f (x) ± g(x)] = A ± B lim in m The limit of a function Dr Lê Xuân Đại (HCMUT-OISP) https://fb.com/sinhvienzonevn THE LIMIT AND CONTINUITY OF A FUNCTION HCM — 2016 / 68 C o Calculating Limits using the limit laws SOLUTION on x3 − 5x + Evaluate lim x→3 x2 − e EXAMPLE 1.1 hV ie nZ (x3 − 5x + 4) x3 − 5x + lim x→3 lim = = x→3 x2 − lim(x2 − 2) in m The limit of a function Dr Lê Xuân Đại (HCMUT-OISP) x→3 33 − 5.3 + 16 = = 32 − https://fb.com/sinhvienzonevn THE LIMIT AND CONTINUITY OF A FUNCTION HCM — 2016 10 / 68 ie nZ on e C o Slant asymptotes lim f (x) = +∞ means that the values of f (x) can be x→+∞ hV made arbitrarily large (larger than M, where M is any positive number) by taking x sufficiently large (larger than N, where N depends on M ) in m Limits involving infinity Asymptotes Dr Lê Xuân Đại (HCMUT-OISP) https://fb.com/sinhvienzonevn THE LIMIT AND CONTINUITY OF A FUNCTION HCM — 2016 54 / 68 Slant asymptotes C o EXAMPLE 3.6 Find lim (x2 − x) e x→+∞ on SOLUTION It would be wrong to write lim (x2 − x) = lim x2 − lim x = ∞ − ∞ x→+∞ nZ x→+∞ x→+∞ ie The Limit Laws can not be applied to infinite limits because ∞ is not a number However, we can write hV lim (x2 − x) = lim x(x − 1) = ∞ × ∞ = ∞ x→+∞ x→+∞ because both x and x − become arbitrarily large and so their product does too in m Limits involving infinity Asymptotes Dr Lê Xuân Đại (HCMUT-OISP) https://fb.com/sinhvienzonevn THE LIMIT AND CONTINUITY OF A FUNCTION HCM — 2016 55 / 68 C o Slant asymptotes e EXAMPLE 3.7 on x2 + x Find lim x→∞ − x ie nZ SOLUTION We divide the numerator and denominator by the highest power of x in the denominator hV x2 + x x+1 ∞ lim = lim = = −∞ x→∞ − x x→∞ − −1 x in m Limits involving infinity Asymptotes Dr Lê Xuân Đại (HCMUT-OISP) https://fb.com/sinhvienzonevn THE LIMIT AND CONTINUITY OF A FUNCTION HCM — 2016 56 / 68 C o Discontinuity at a number e DEFINITION 4.1 Function f (x) is discontinuous at a if f (x) is not continuous at a nZ on If f (x) is discontinuous at a then at least one of the following equalities is not true lim f (x) = lim f (x) = f (a) x→a+ x→a− hV ie This means that: at least one of the following limits lim f (x) and x→a+ lim f (x) does not exist or is equal ∞ x→a− both limits lim f (x) and lim f (x) exist but at least x→a+ x→a− one of the above equalities is not true https://fb.com/sinhvienzonevn in m Discontinuity Dr Lê Xuân Đại (HCMUT-OISP) THE LIMIT AND CONTINUITY OF A FUNCTION HCM — 2016 57 / 68 C o Removable discontinuity x→a− hV ie nZ x→a+ on e DEFINITION 4.2 A function f has a removable discontinuity at a if lim f (x) and lim f (x) exist and either f (a) is undefined x→a+ x→a− or lim f (x) = lim f (x) = f (a) (16) in m Discontinuity Dr Lê Xuân Đại (HCMUT-OISP) https://fb.com/sinhvienzonevn THE LIMIT AND CONTINUITY OF A FUNCTION HCM — 2016 58 / 68 C o Removable discontinuity |x|, 1, x=0 x=0 on f (x) = e EXAMPLE 4.1 Find all discontinuities of function nZ SOLUTION lim f (x) = lim |x| = lim x = ie x→0+ x→0+ x→0+ hV lim f (x) = lim |x| = lim (−x) = x→0− x→0− x→0− So lim f (x) = lim f (x) = = = f (0) Therefore, f has a x→0+ x→0− removable discontinuity at a = in m Discontinuity Dr Lê Xuân Đại (HCMUT-OISP) https://fb.com/sinhvienzonevn THE LIMIT AND CONTINUITY OF A FUNCTION HCM — 2016 59 / 68 C o Jump discontinuity e DEFINITION 4.3 A function f has a jump discontinuity at a if lim f (x) x→a+ and lim f (x) exist and on x→a− (17) lim f (x) = lim f (x) x→a− hV ie nZ x→a+ in m Discontinuity Dr Lê Xuân Đại (HCMUT-OISP) https://fb.com/sinhvienzonevn THE LIMIT AND CONTINUITY OF A FUNCTION HCM — 2016 60 / 68 C o Jump discontinuity e EXAMPLE 4.2 Find all discontinuities of function x>0 x=0 x
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