explain why the function is discontinuous at the given point ( f(x) = { x^2-2x-8/ x-4 if x isnt equal to 4 and 3 if x=4 .. ( is it because there is a hole at 4 ?)

Question

explain why the function is discontinuous at the given point ( f(x) = { x^2-2x-8/ x-4  if x isnt equal to 4 and 3 if x=4 .. ( is it because there is a hole at 4 ?)

zarkon · Answer

$$f(4)=3$$

but

$$\lim_{x\to 4}\frac{x^2-2x-8}{x-4}=6$$

aravindg · Answer

satellite help me amistre could not answer my question

anonymous · Answer

it isn't equal to anything at 4 because you will have 
$$4-4=0$$ in the denominator

zarkon · Answer

she has a piecewise function

aravindg · Answer

satellite help me amistre could not answer my question

aravindg · Answer

satellite help me amistre could not answer my question

anonymous · Answer

it is true you can rewrite by factoring and canceling and gettin 
$$f(x)=x+2$$ but that is only if 
$$x\neq4$$

anonymous · Answer

oh i should learn to read hold on

liizzyliizz · Answer

yeah it is x$$\neq4$$ lol

anonymous · Answer

$$f(x) = \left\{\begin{array}{rcc}
\frac{x^2-2x-8}{ x-4} & \text{if}  & x \neq 4 \
3& \text{if}  & x = 4
\end{array}
\right.
$$

liizzyliizz · Answer

yess

anonymous · Answer

ooooh ok

anonymous · Answer

so 
$$\frac{x^2-2x-8}{x-4}=\frac{(x-4)(x+2)}{x-4}=x+2$$ which holds if 
$$x\neq 4$$

anonymous · Answer

so this function, even though it looks like a rational function, is just the line 
$$y=x+2$$ with a hole in it where (4,6) should be right?

aravindg · Answer

satellite help me after this???

anonymous · Answer

sure hold on

liizzyliizz · Answer

so i was right it was just the hole..

anonymous · Answer

meaning that if you wanted to "remove" the discontinuity, you would just call it 
$$f(x)=x+2$$ and be done

anonymous · Answer

or if you want to spend a lot of ink you could write 
$$f(x) = \left\{\begin{array}{rcc}
\frac{x^2-2x-8}{ x-4} & \text{if}  & x \neq 4 \
6 & \text{if}  & x = 4
\end{array}
\right.$$

anonymous · Answer

because the function really wants to be 6 when x = 4.  but some idiot decided to call it 3 there, so it is not continuous

liizzyliizz · Answer

lol the idiot would be my math book -_-

anonymous · Answer

and yes, you are right, it is a "hole" which is sometimes called a 'removable discontinuity" because if you define the function correctly you an "remove" it

liizzyliizz · Answer

yeah. ahh ok one last question my next problems involve using theorems (4,5,7, and 9) im not quite sure what that is..  for some functions.. x_x is it something my hw didnt specify?

aravindg · Answer

come