Removable vs nonremovable discontinuities with absolute values and trig functions

[EasyStyle4747]

Help on continuity!

1) Ok, i know how to find x, but how do you know if something is nonremovable or removable discontinuity?
like for this:

f(x)=|x+2|/(x+2)

i knoe its x=-2, but is it nonremovable or removable?

2) How do u do the continuity stuff with there:

f(x)=csc2x

f(x)=tan(pi*x/2)

Plz try to explain in simple terms.

 

[vsage]

This is just to get the ball rolling but think about lim(x->-2+) |x+2|/(x+2) and lim(x->-2-) |x+2| / (x+2).

 

[HallsofIvy]

1) Ok, i know how to find x, but how do you know if something is nonremovable or removable discontinuity?
like for this:

f(x)=|x+2|/(x+2)

i knoe its x=-2, but is it nonremovable or removable?
[/quote]
Knowing the definition would be a good start. What is the definition of "removable discontinuity"?

2) How do u do the continuity stuff with there:

f(x)=csc2x

f(x)=tan(pi*x/2)

Plz try to explain in simple terms.

What do YOU mean by "do the continuity stuff"?

 

[shirewolfe]

I'm not an expert in this but perhaps the following explanation may help.

Removeable discontinuity occurs when the discontinuous 'point' or gap can be redefined to make the graph continuous. Such as if f(x)=x occurs but is undefined at x=3 (possibly due to a set interval). If you define f(x) at x=3 as 3 (as the regular function f(x)=x woild pass through point (3,3) you could 'remove' the discontinuity from the graph.

As another explanation, removeable discontinuity generally occurs when a limit at the discontinuous 'point' exists, meaning that
lim f(x) = lim f (x)
x->c+ x->c-
and furthermore that f(x) approaches a definite real number as x approaches c ( F(x) should not be appreaching + or - infinity.

Non-removable discontinuity exists when the discontinuous 'point' in the function cannot be redefined to make the graph continuous. This occurs mainly when a major gap exists as the discontinuity, when
lim f(x) and lim f (x) are not equal, or
x->c+ x->c-
in other words, f(x) does not approach the same limit when c is approached from the left and when c is approached from the right.

I don't know how efficient my explanation is, but i hope it helps.