Approaching the Limit of a Sin Function Using Calculus Techniques

[Jimbo57]

Homework Statement

[PLAIN]http://www4d.wolframalpha.com/Calculate/MSP/MSP120619i0ea2h15bci31300005d9a3422292h43h4?MSPStoreType=image/gif&s=43&w=174&h=39

Homework Equations

The Attempt at a Solution

So Wolfram says to use L'Hopital as the first step, we haven't learned anything about this yet so there has to be another way using calculus that a first year math student would know.

I would first expand the denominator of x^2-4 to (x+2)(x-2) and cancel out the x+2 in both numerator and denominator. And I'm left with:
((x-1)sin1)/(x-2)
This is where I get lost...

Any help would be much appreciated!
Jim

 

[grzz]

One cannot separate (x+2) from sin any more than one can separate ... a bone from the mouth of a very hungry dog!

Either one 'cancels' the whole sin(x+2) or nothing at all.

 

[LCKurtz]

You don't cancel the (x+2)'s. You should know

$$\lim_{x\rightarrow 0}\frac{\sin x}{x}$$

Use that somehow.

 

[grzz]

Did you mention in class that lim(a x b) = lim(a) x lim(b)?

 

[Jimbo57]

Yes, we mentioned all the above, and this was much easier after LCKurtz's suggestion:

=lim sin((x+2)/(x+2)) * lim (x-1)/(x-2)
= 1 * -3/-4
=3/4
I really need to learn how to use that equation generator thing...

Much appreciated folks! :)