Calculating Limits Involving Cosine: Is My Approach Correct?

[mtayab1994]

Homework Statement

find the limit of :

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

Homework Equations

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?

 

[tiny-tim]

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!

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.

 

[tiny-tim]

ah!

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

you'll learn about that later ​

 

[mtayab1994]

ah!

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

you'll learn about that later ​

Alrighty thank you very much. I live in morocco and I'm in kind of like in a "Math-Science" major in high school so we're really big on math and physics over here.