Singularities or discontinuities

• May 26th 2009, 01:54 PM
NaNa
Singularities or discontinuities
ok I have understand how to do it but i ave confuse when is removable and when is non-removable

here is the question:
Find any singularities or discontinuities in the following functions:

f(x)=(x+3)/(x^2-9)
(an:non-removable singularity at x=3, removable singularity at x=-3, put f(x)=-1/6)

f(x)=cosx/(x-π/2)
(an:removabe singularity at x=π/2, put f(π/2)=-1)
• May 26th 2009, 02:03 PM
Calculus26
A discontinuity is removable if the limit exists

Remember Continuity means lim x->a f(x) = f(a)

which implies
1. f(a) is defined
2. lim x->a f(x)
3. 1 = 2

For example

f(x)=(x+3)/(x^2-9)

limas x->-3 = -1/6 so if we define f(-3) =-1/6 all conditions are met

limasx->3 DNE fails 2 cannot remove discontinuity
• May 26th 2009, 02:15 PM
NaNa
why the lim x->3= -1/6!???!?i found it -infinite
• May 26th 2009, 02:18 PM
Calculus26
Re read the post I said lim x-> -3 = -1/6

and lim x-> 3 DNE
• May 26th 2009, 02:18 PM
TheEmptySet
Quote:

Originally Posted by NaNa
ok I have understand how to do it but i ave confuse when is removable and when is non-removable

here is the question:
Find any singularities or discontinuities in the following functions:

f(x)=(x+3)/(x^2-9)
(an:non-removable singularity at x=3, removable singularity at x=-3, put f(x)=-1/6)

f(x)=cosx/(x-π/2)
(an:removabe singularity at x=π/2, put f(π/2)=-1)

The simple idea is that if $\displaystyle f(x)$ is undefined at a point $\displaystyle x_0$ but the limit at the point $\displaystyle \lim_{x \to x_0}f(x)=c < \infty$ the we can extend the function to the value c at $\displaystyle x_0$ ie $\displaystyle f(x_0)=c$

Edit: Geez I am really late :(
• May 26th 2009, 02:21 PM
NaNa