alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

# So we can see that as x tends irr ir to a the

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ientl y c lose means as c lose as necessar y . a s close If we want 0 < C f(x)− 2 aC < ε some discrete infintesimal as small as we like, we can choose x sufficiently close to a. For example, we may choose : ε 0 < C x− a C < δ , w here δ = − 2 Work it out: ε ε ε 0 < C x− a C < δ ⇒ 0 < C x− a C < − ⇒ a −− < x < a + − 2 2 2 ε ) + a = 2a + − ε f(x) = x + a = ( a + − -- 2 -- 2 ε ε Now: 0 < C f(x) --2a C = C (2a + − ) -- 2a C = − which is < ε -- 2 2 ε So when C x -- a C < − we will have C f(x) -- 2a C < ε 2 We can say: Limit f(x) = 2a x→a Instead of saying: Limit f(x) which is Limit (x + a) x→a x→a we can use the infinitesimal δ x and say: Limit (x + a) = L imit ((a-- δ x ) + a ) = 2a x → a− δx → 0 and Limit (x+a) = L imit ((a+ δ x)+a) = 2a x → a+ δx → 0 35 (from the left ) left lef (from the right) right Let us now look at the example 9 of the previous chapter. Example 9 : +1 if x is rational rational f(x) = { 0 if x is ir r ational irr ir Limit f(x) = b . Let us try to choose an ε such that C f(x) --bC > ε Suppose the x → a for some x very close to a, that is to say for 0 < C x − a C < δ . Recall what we said about the rationals and irrationals. Between any two rationals there are infinitely many rationals. And between any two rationals there are also rationals irr infinitely many ir r ationals . ir No matter how close x is to a there will always be infinitely many rational rational numbers and infinitely many ir r ational numbers. So we can see that as x TENDS irr ir TO a, the function f(x) will keep fluctuating : f(x) will be 0 or 1. There are two possibilities : b = 0 or b ≠ 0. Case b = 0 : choose ε = 1/2 No matter how close x is to a , that is to say no matter how small the δ w e choose, there will always be some r a tional n umbers such that 0 < C x− aC < δ . If x is rational then C f(x) --bC = C 1 --0C = 1 > ε = 1/2 . So b = 0 cannot...
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern