show that f(x) = {x 2 - 3x + 2 / x - 2, if x does not equal to1 {1 , if x = 1 is

Question

show that

f(x) = {x2 - 3x + 2 / x - 2, if x does not equal to1
         {1                         , if x = 1

is discontinuous at x = 1

what i did was factor out the equation like so and find thelimit
limx->1 (x-2)(x-1) / (x-2), then cancel out thecommon terms
lim x->1 x-1 = 0

and do the limit the answer was 0. in the solutions manual howeverafter factoring the limit was
lim x->1 1/x-1=

i don't understand why it is (1/x-1) and not just x-1, the 1 isonly over something if all the terms from the numerator arecanceled out...right?

Explanation / Answer

I agree this is an odd question - it would make more sense if youwere given f(x) = (x-2) / (x2 - 3x + 2). As you (indirectly) point out, that would equate to(x-2)/(x-2)(x-1) (if x does not equal 1). That would then cancel out to 1/(x-1). The limit of 1/(x-1)as x approaches 1 becomes infinity because you are approaching 1/0,which is undefined. Specifically, approaching x = 1 from theright, with a vertical asymptote at x = 1, the graph lookssomething like the spread of your index finger and thumb. Oran L if you want to be crude. And you're rocketing upwardtoward infinity at x ~ 1. In short, either the book or the answer (likely the book) reversedthe equation.

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