{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

X 0 x 0 x0 value fx value x0 x0 sinx x

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

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: : near 0 and to the left of 0 : x = 0 − δx left sin(0 − δ x) − δx sin(− δ x) }={ }=1 } ={ f(x) = { (0 − δ x) − δx − δx Note how we simplify ffirst and then take the limit. We can simplify first because irst take simplify ir δx only TENDS TO zero. δx ≠ 0. There is no division by zero in this step. here division by zer ero step 2. LIMIT step : L imit {1} = 1 δx → 0 f(x) = − 4π − 3π − 2π − 1π 0 +1π sin (x) x +2π +3π +4π . . . x * Note: we may infer this from the basic definition of sin (θ) = opposite side / hypoteneuse. When θ is infinitely small, say δ θ , then sin ( δ θ ) = r δ θ/r = δ θ . Another way is to look at the expansion of sin(x). When x becomes infinitely small, say δ x , then only the first term in the expansion matters. 32 Limit from the RIGHT 1. TENDS TO step : near 0 and to the right of 0 : x = 0 + δx right δx sin(0 + δ x) sin(+ δ x) } =1 }= { }={ f(x) = { δx (0 + δ x) + δx Note how we simplify ffirst and then take the limit. We can simplify first because irst take simplify ir δx only TENDS TO zero. δx ≠ 0. There is no division by zero in this step. here division by zer ero step 2. LIMIT step : L imit δx → 0 {1} = 1 sin(x) imit imit Since, L → -- { f(x) } = 1 = L→ + { f(x) } , we say L imit { x } = 1 . x 0 x 0 x→0 Value { f(x) } = Value x=0 x=0 sin(x) { x } = 0 0 , which is something undefined. / Value Example 9 : We now present an example of a function f(x) where x = a f(x) L imit exists everywhere (i.e. a can be any real number), but the x → a f(x) does NOT exist anywhere. f(x) = +1 if x is rational rational { 0 if x is irrational irr ir 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 infinitely many irrationals. So we can see that as x TENDS TO a, the function f(x) will keep fluctuating. The function f(x) will be either 0 or 1. It will not take on a single specific value. Hence we can say that the Limit f(x) does NOT exist. Since a can x→a be any point on the real number line, the L imit does NOT exist anywhere. On fi...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    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.

    Student Picture

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

  • Left Quote Icon

    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.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    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.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern