How to Solve a Limit Calculation Problem with Multiple Choice Options?

[nick850]

Homework Statement

This is a problem off a multiple choice practice test:
lim t->3 ( 1/(t^2-3t) - 2/(t^2-9) =

The solutions are:
(a) 0 (b) -1/9 (c) -1/18 (d)1 (e) 1/3

The correct answer is (c).

Can someone explain to me how to solve it? Any help would be very appreciated. You don't need to explain how to do limits. I just need to know how to manipulate the equation so that it's not indeterminate.

Homework Equations

NA

The Attempt at a Solution

I tried multiplying by the conjugate with no success.

 

[HallsofIvy]

The first thing I would do is actually subtract the two fractions:
$$\frac{1}{t^2- 3t}- \frac{2}{t^2- 9}= \frac{1}{t(t- 3)}- \frac{2}{(t- 3)(t+ 3)}$$

Clearly the "common denominator" is t(t- 3)(t+ 3):
$$\frac{t+ 3}{t(t- 3)(t+ 3)}- \frac{2t}{t(t- 3)(t+ 3)}$$[tex]= \frac{t+ 3- 2t}{t(t- 3)(t+ 3)}= \frac{-t+ 3}{t(t- 3)(t+ 3)}= -\frac{t- 3}{t(t- 3)(t+ 3)}[/itex][/tex]

 

[nick850]

Thank you!