Is the Limit as x Approaches 2 of ((1/x)-(1/2))/(x-2) Zero?

[TayTayDatDude]

Homework Statement

lim
x\rightarrow2 ((1/x)-(1/2))/(x-2)

Homework Equations

The Attempt at a Solution

Limit x-> for all of them, too tedious to keep rewriting it
=((2/2x)-(x-2x))/(x-2)
=((2-x)/(2x))/(x-2)
=((2-x)/(2x)) * (1/(x-2))
=(2-x)/(2x^2-4x)
sub 2 in = 0/0 = 0

therefore when x approaches 2, y approaches 0?

=

 

[Dick]

Never say 0/0=0. Never, until you have actually shown it. You have (2-x)/((2x)*(x-2)). (2-x)/(x-2)=-(x-2)/(x-2)=(-1). Now what's the limit?

 

[TayTayDatDude]

Never say 0/0=0. Never, until you have actually shown it. You have (2-x)/((2x)*(x-2)). (2-x)/(x-2)=-(x-2)/(x-2)=(-1). Now what's the limit?

How did you go from (2-x)/((2x)*(x-2)) to (2-x)/(x-2)?

 

[Dick]

I didn't. I just separated (2-x)/((2x)*(x-2)) into [1/(2x)] * [(2-x)/(x-2)] and made the observation that the second factor is -1.

 

[TayTayDatDude]

I didn't. I just separated (2-x)/((2x)*(x-2)) into [1/(2x)] * [(2-x)/(x-2)] and made the observation that the second factor is -1.

So then it is 0?

Btw, does the derivative of x/(2x-1) = 0?

 

[Dick]

So then it is 0?

Btw, does the derivative of x/(2x-1) = 0?

Is what 0? No, the derivative of x/(2x-1) is not zero. Unless x=0.