Question

f(x)= ((x-4)(x+5)/((13x+4)(4-x))

What graphical feature occurs at x=4? (select all that apply)
hole
intercept
vertical asymptote

Answer 1

\[f(x) = \frac{(x-4)(x+5)}{(x-4)(13x+4)}\]
This function contains a removeable discontinuity.
Obviously if you set x = 4 you get 0/0 which is undefined but since you have (x-4) in both numerator and denominator you can remove those (they cancel out), just remember that the function is undefined for x = 4.

So your f(x) when x is really close to 4 can be calculates as if x is = 4 (except it cannot be exactly 4)
\[f(x) = \lim_{x \rightarrow 4} \frac{(x+5)}{(13x+4)} = \frac{9}{56}\]

so f(x) is 9/56 when x is really really close to 4, so what is the behavior when x is 4 ?

Answer 2

hole

Answer 3

yep

Answer 4

Just a minor thing but the lim should be
\[-\frac{ 9 }{56 }\]
if the numerator contains (x-4) and the denominator contains (4-x).

Answer 5

oops - sorry, my mistake, I didnt notice different terms. As you can see I have both numerator and denominator as (x-4). But you are right, one needs to be multiplied by -1. Thank you for pointing this out.

Answer 6

Just a minor thing but the lim should be