# Limits of fractions of polynomials and trig functions

1. ### carbz

35
I have two...

1. The problem statement, all variables and given/known data
The the limit

2. Relevant equations
$\lim_{x \rightarrow 1} \frac{1-cosx}{x^2}$

3. The attempt at a solution
I figured to just plug in 1, but I wanted to make sure....

1. The problem statement, all variables and given/known data
Find the limit

2. Relevant equations
$\lim_{x \rightarrow 3} \frac{\sqrt{x^2-6x+9}}{x-3}$

3. The attempt at a solution
I plugged in the 3, and got 3/0, then I got lost...

2. ### EnumaElish

2,481
In 2, did you try simplifying the numerator? (What are the roots of the polynomial?)

3. ### carbz

35
Yes, I tried doing that.

$(x-3)(x-3)$

However, I forgot how to get rid of that radical. Squaring wouldn't work, so I have no idea.

Also, no thoughts on the first one?

4. ### EnumaElish

2,481
What is a short hand expression for (x-3)(x-3)?

5. ### carbz

35
(x-3}^2. Oh yeah, so that takes away the square root, and after everything, it leaves 0. thank you.

6. ### Avodyne

1,270
In the first one, are you sure the problem isn't x->0 instead of x->1 ?

7. ### carbz

35
it is 1, not 0.

8. ### arildno

12,015
Well, then your book has a typo..

9. ### carbz

35
it's not from my book, it was my teacher.

10. ### Avodyne

1,270
Well, it's 99% certain that your teacher meant to write 0 instead of 1. With 1, it's trivial, since both the numerator and denominator are finite, nonzero constants in that limit.

11. ### arildno

12,015
Then he either blundered, or tried to fool you.

Your function is defined&continuous on all values of x except x=0.

Your original approach is perfetly valid in the case of x=1.

12. ### carbz

35
allright, thankyou.

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