Limits of fractions of polynomials and trig functions

[carbz]

I have two...

Homework Statement

The the limit

Homework Equations

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

The Attempt at a Solution

I figured to just plug in 1, but I wanted to make sure...

Homework Statement

Find the limit

Homework Equations

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

The Attempt at a Solution

I plugged in the 3, and got 3/0, then I got lost...

 

[EnumaElish]

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

 

[carbz]

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?

 

[EnumaElish]

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

 

[carbz]

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

 

[Avodyne]

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

 

[carbz]

it is 1, not 0.

 

[arildno]

Well, then your book has a typo..

 

[carbz]

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

 

[Avodyne]

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.

 

[arildno]

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.

 

[carbz]

allright, thankyou.