How to solve this limit?

Question

lim as x tends to 2 [sin(2x-4)]/(x-2) . 

 

I attempted to solve but im probably wrong. I factored out the 2x-4 then canceled the x-2 with the x-2 from the factored numerator. But then im just left with sin 2 ?

Answer 1

Start with

 

2 [sin(2x-4)] / (x-2)

 

As x→2, you approach 0/0, which is an indeterminate form.  Use L'Hopital's rule to solve for an indeterminate form.  Are you familiar with it?  If not, let me know and I can explain it further.  For now, let us apply the rule:

 

limit as x→2 of 2 [sin(2x-4)] / (x-2) = limit as x→2 of 4 [cos(2x-4) / 1 = 4 cos(0) = 4(1) = 4

 

The answer is 4.

Answer 2

let A = (x - 2)

 

then your limit becomes as A --> 0

because as x --> 2 , A --> 0

 

then we have:

 

Lim 2 sin(2A) / A

A-->0

 

Multiply the top and bottom by 2 and you have the standard Limit sinx/x = 1 as x -->0

 

Lim 4 sin(2A) / 2A = 4

A -->0