Calculating Limits Involving Cosine: Is My Approach Correct?

In summary, the conversation discussed finding the limit of a function involving cos(x) and x^2, and the use of a conjugate and a trigonometric identity to simplify the expression. The final result was determined to be 1/8, and the conversation also touched on the concept of replacing cos(x) with 1-x^2/2.
  • #1
mtayab1994
584
0

Homework Statement


find the limit of :

[tex]\lim_{x\rightarrow0}\frac{\sqrt{5-cos(x)}-2}{x^{2}}[/tex]


Homework Equations


The Attempt at a Solution



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

[tex]\frac{1-cos(x)}{x^{2}(\sqrt{5-cos(x)}+2)}[/tex]

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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  • #2
hi mtayab1994! :smile:
mtayab1994 said:
I multiplied the numerator and the denominator by the conjugate of the numerator and i got :

[tex]\frac{1-cos(x)}{x^{2}(\sqrt{5-cos(x)}+2)}[/tex]

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! :-p

yes, that's fine

(the usual way of dealing with cosx is to replace it by 1 - x2/2 … same result)
 
  • #3
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.
 
  • #4
ah!

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

you'll learn about that later :smile:​
 
  • #5
tiny-tim said:
ah!

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

you'll learn about that later :smile:​

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.
 

FAQ: Calculating Limits Involving Cosine: Is My Approach Correct?

What is the definition of a limit involving cosine?

The limit of a function involving cosine is the value that the function approaches as the input approaches a specific value, known as the limit point. This can be represented using mathematical notation as lim f(x) = L, where x approaches the limit point and L is the limit value.

How do you evaluate a limit involving cosine?

To evaluate a limit involving cosine, you can use algebraic manipulation, trigonometric identities, and the limit laws. You can also use a graphing calculator or a table of values to estimate the limit value.

What are the common types of limits involving cosine?

There are three common types of limits involving cosine: limits at a finite point, limits at infinity, and limits involving trigonometric identities. Limits at a finite point involve finding the limit of a function as the input approaches a specific value. Limits at infinity involve finding the limit of a function as the input approaches positive or negative infinity. Limits involving trigonometric identities involve using trigonometric identities to simplify the function and then evaluating the limit.

How do you solve a limit involving cosine that is indeterminate?

If a limit involving cosine results in an indeterminate form, such as 0/0 or ∞/∞, you can use L'Hôpital's rule to solve it. This rule states that if the limit of two functions is indeterminate, then the limit of their derivatives will be the same as the original limit. By taking the derivative of the numerator and denominator separately, you can often simplify the function and evaluate the limit.

What are the applications of limits involving cosine in real life?

Limits involving cosine have various applications in physics, engineering, and other fields. One example is in calculating the acceleration of a moving object by taking the limit of its velocity as time approaches zero. This is known as finding the instantaneous rate of change. Other applications include determining the maximum height of a projectile or finding the displacement of a vibrating object.

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