Engineering Calculus Notes 231

Engineering Calculus Notes 231 - it has a limit along some...

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

View Full Document Right Arrow Icon
3.1. CONTINUITY AND LIMITS 219 corresponding sequence of values of f ( x 0 ) converges to L : −→ x 0 negationslash = −→ x k −→ x 0 f ( −→ x k ) L. The same arguments that worked before show that a function converges to at most one number at any given point, so we can speak of “the” limit of the function at −→ x = −→ x 0 , denoted L = lim −→ x −→ x 0 f ( −→ x ) . For functions of one variable, we could consider “one-sided” limits, and this often helped us understand (ordinary, two-sided) limits. Of course, this idea does not really work for functions of more than one variable, since the “right” and “left” sides of a point in the plane or space don’t make much sense. We might be tempted instead to probe the limit of a function at a point in the plane by considering what happens along a line through the point: that is, we might think that a function has a limit at a point if
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: it has a limit along some line (or even every line) through the point. The following example shows the folly of this point of view: consider the function de±ned for −→ x n = −→ ∈ R 2 by f ( x,y ) = xy x 2 + y 2 , ( x,y ) n = (0 , 0) If we look at the values of the function along a line through −→ x = −→ 0 of slope m , y = mx, we see that the values of f ( −→ x ) at points on this line are f ( x,mx ) = ( x )( mx ) x 2 + m 2 x 2 = m 1 + m 2 . This shows that for any sequence of points approaching the origin along a given line, the corresponding values of f ( −→ x ) converge—but the limit they converge to varies with the (slope of) the line , so the limit lim −→ x → −→ x f ( −→ x ) does not exist....
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