how do you simply the limit algebraically:

lim as x-0 (x^2)/(1-cos(x))

Question

how do you simply the limit algebraically:

lim as x->0 (x^2)/(1-cos(x))

anonymous · Answer

answer please!!!!

blockcolder · Answer

$$\lim_{x \to 0} \frac{x^2}{1-\cos x}\cdot \frac{1+\cos x}{1+\cos x}=\lim_{x \to 0} \frac{x^2}{1-\cos^2x}\cdot \lim_{x \to 0}(1+\cos x)=1\cdot 2=2$$

anonymous · Answer

im starting to understand how you solve these, but how do you know the method that works? sometimes i try one way and the answer turns out to 0 and other ways it turns out to a real number (like this one), if i get the answer 0 should i just try the question another way to see if i get a different answer?

blockcolder · Answer

You get that with experience. :)

anonymous · Answer

alright, thank you!

anonymous · Answer

my friend told me to use the "L-hospital rule", would that be a viable method of doing these questions? The rule is to take the derivative of these functions.

blockcolder · Answer

Yes, but you have to make sure that you get 0/0 before doing it, and that there is no other way of doing it.

anonymous · Answer

0/0 as in when i just simply plug the lim x-> 0 in?

blockcolder · Answer

Yeah, like in this one.

anonymous · Answer

alright, thank you again!

anonymous · Answer

Apply L'Hospital's rule.
Taking the derivative of the numerator over the derivative of the denominator twice
$$
\frac {2x} {\sin (x)}\
\frac 2 {\cos(x)}

$$
The limit of the second fraction is 2/1=2