# Limits involving cosine

mtayab1994

## Homework Statement

find the limit of :

$$\lim_{x\rightarrow0}\frac{\sqrt{5-cos(x)}-2}{x^{2}}$$

## The Attempt at a Solution

I multiplied the numerator and the denominator by the conjugate of the numerator and i got :

$$\frac{1-cos(x)}{x^{2}(\sqrt{5-cos(x)}+2)}$$

then: i divided top and bottom by x^2 and i got (1/2)/4 which is 1/8.

Is what I have done correct?

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Homework Helper
hi mtayab1994!
I multiplied the numerator and the denominator by the conjugate of the numerator and i got :

$$\frac{1-cos(x)}{x^{2}(\sqrt{5-cos(x)}+2)}$$

then: i divided top and bottom by x and i got (1/2)/4 which is 1/8.

Is what I have done correct?

oooh, I've never seen that trick before! :tongue2:

yes, that's fine

(the usual way of dealing with cosx is to replace it by 1 - x2/2 … same result)

mtayab1994
Yea thanks. And by the way I didn't know that you replace cos x with 1-x^2/2. Well maybe it's because we haven't done it yet.

Homework Helper
ah!

yes, cos x = 1 - x2/2 + x4/4! - x6/6! + …