Limit of cos x as x approaches 1 formal

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In summary, the conversation involves a student seeking help with finding the limit of Cos X as x approaches one using the formal definition. They have attempted to solve the problem using epsilon values of .1, .001, and .00001 and have set up an inequality for each value. They also suggest using the identity \cos{a} - \cos{b} = -2 \sin \frac {a+b} 2 \sin \frac {a-b} 2 to help solve the problem.
  • #1
enkerecz
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Homework Statement



The limit of Cos X as x approaches one using the formal definition.
Given epsilon values of .1, .001, .00001


Homework Equations





The Attempt at a Solution



Spent all period in physics calc trying to solve this using the formal definition.
Since there is no way to manipulate abs(cosx-.5403)<epsilon
I proceeded to make my delta = the epsilon values and plug into the equation.
None of the answers were working correctly. Any help would be greatly appreciated because i have a feeling that I am missing something here.
 
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  • #2
It appears you want to find the individual values of delta when epsilon is .1, .001, and .00001.

Set up your inequality for each of these values.
For epsilon = .1

|cos x - 0.5403023| < .1
<==> -.1 < cos x - 0.5403023 < .1
Continue in this vein until you have cos x between two values, and then use cos-1 to get an inequality in x. At that point you should have a good idea what to use for delta.

Do the same for the other two values of epsilon.
 
  • #3
Hint: You might find this identity useful:

[tex]\cos{a} - \cos{b} = -2 \sin \frac {a+b} 2 \sin \frac {a-b} 2[/tex]
 

FAQ: Limit of cos x as x approaches 1 formal

What is the limit of cos x as x approaches 1 formally?

The limit of cos x as x approaches 1 formally is equal to cos 1. This means that as x gets closer and closer to 1, the value of cos x will approach the value of cos 1, which is approximately 0.5403.

How do you calculate the limit of cos x as x approaches 1?

To calculate the limit of cos x as x approaches 1, you can use the formal definition of a limit: lim x→a f(x) = L. In this case, a is equal to 1 and f(x) is equal to cos x. Substitute these values into the definition and solve for L to find the limit.

Why is the limit of cos x as x approaches 1 important?

The limit of cos x as x approaches 1 is important because it helps us understand the behavior of the cosine function near the value of 1. It also allows us to make predictions about the value of cos x at x = 1 without having to actually evaluate the function at that point.

Can the limit of cos x as x approaches 1 be different from cos 1?

No, the limit of cos x as x approaches 1 cannot be different from cos 1. This is because the limit represents the value that the function approaches as x gets closer and closer to 1, while cos 1 is the actual value of the function at x = 1. In this case, both the limit and the function value are equal to cos 1.

How does the graph of cos x approach the value of cos 1 as x gets closer to 1?

The graph of cos x approaches the value of cos 1 as x gets closer to 1 by getting closer and closer to the x-axis. This can be seen by plotting points on the graph for values of x that are very close to 1. As x approaches 1, the points on the graph will get closer and closer to the point (1, cos 1), which is the value of cos 1 on the x-axis.

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