What is the answer to this limit problem?

Question

Lim x-->2- (x approaches 2 from the left) 2x-4/x-2

What if the limit of the numerator and denominator is both 0?

Answer 1

Lim x-->2- (x approaches 2 from the left) 2x-4/x-2
the answer is 2

you can factor out a 2 from the numerator
2(x-2) / (x-2)

the (x-2) cancel out, and your left with 2

when you take the limit of a constant number the answer is that number

if the limit of the numerator and denominator is both 0 then your limit is undefined

Answer 2

you know (2x-4)/(x-2) = 2 for all x <> 2, so the graph is y=2 with a hole at x = 2. l'Hopital's rule says when the limit is 0/0 find the derivatives of both parts of the fraction, which gives us 2/1 = 2.

Answer 3

I rewrote the question to what I think may have been meant:

lim(x->2)(x_left)(( 2x-4)/(x-2))
= lim(x->2)(x_left)(( 2)/(1))
= 2 whether you take the limit from the left, or from the right, or both.

Answer 4

In this case you'll need to differentiate top and bottom with respect to x and then take limits :)