Question

Problem in replies. Limits.

Answer 1

Suppose \[\lim_{x \rightarrow -3^-}f(x)=-1, \lim_{x \rightarrow -3^+}f(x)=-1\] and f(-3) is not defined. Which of the following statements is (are) true.
I. lim x->-3 f(x)=-1
II. f is continuous everywhere except at x=-3
III. f has a removable discontinuity at x=-3

Answer 2

A limit at the point x = a exists if the left handed and right handed limits exist at a and are both equal to a finite value L.

For the function to be continuous at x = a, the limit must exist at x=a and equal to the function evaluated at a. i.e. we require f(a) = L. This means the function must also exist as x = a for the continuity at the point.

So ask yourself: does the limit exist at the point?
Then, is the function defined and equal to the limit at the point? If not, then we have a discontinuity.

Answer 3

Further if the limit does exist at the point, but the point is not defined, such a discontinuity is called a removable discontinuity - a gap that could be filled in by redefining the function.

If the limit does not exist there, we have a jump discontinuity or one towards infinity

http://www.mathwarehouse.com/calculus/continuity/what-are-types-of-discontinuities.php

Answer 4

Thanks for the help!