Solve Limit as x→π/8: (cos(2x)-√(2))/(x-π/8)

[batman2002]

Homework Statement

Find the limit as x approaches ∏/8, (cos(2x)-√(2))/(x-∏/8)

Homework Equations

cos(2x)+cos(2a)

The Attempt at a Solution

I tried to multiply the conjugate of the terms but ended up stuck there, don't know how to go on. Please help.

 

[Mark44]

Homework Statement

Find the limit as x approaches ∏/8, (cos(2x)-√(2))/(x-∏/8)

Homework Equations

cos(2x)+cos(2a)

This isn't an equation, and I don't see how it's relevant to anything.

The Attempt at a Solution

I tried to multiply the conjugate of the terms but ended up stuck there, don't know how to go on. Please help.

As x approaches $\pi$/8, what does the numerator approach? What does the denominator approach?

 

[batman2002]

This isn't an equation, and I don't see how it's relevant to anything.As x approaches $\pi$/8, what does the numerator approach? What does the denominator approach?

You end up with -(1/sqrt2)/0 limit. the equation is an identity that is supposed to help when solving the question.

I also tried expanding the relevant equation and ended up with, cos(2x)+cos(2a)=-2sin(x+a)sin(x-a)

 

[Mark44]

You end up with -(1/sqrt2)/0 limit. the equation is an identity that is supposed to help when solving the question.

cos(2x)+cos(2a) is NOT an equation, so it can't possibly be an identity.

You end up with -(1/sqrt2)/0 limit.

But that's not a number. I agree that the numerator approaches -1/sqrt(2), which is the same as -sqrt(2)/2. And I agree that the denominator approaches 0.

So this problem is similar to these limits:

$$\lim_{x \to 0}\frac{1}{x}$$
$$\lim_{x \to 0}\frac{1}{x^2}$$

How would you characterize these two? (One of them has a direct bearing on your limit.)

 

[batman2002]

I am not exactly sure about that.

cos(2x)+cos(2a) is NOT an equation, so it can't possibly be an identity.


But that's not a number. I agree that the numerator approaches -1/sqrt(2), which is the same as -sqrt(2)/2. And I agree that the denominator approaches 0.

So this problem is similar to these limits:

$$\lim_{x \to 0}\frac{1}{x}$$
$$\lim_{x \to 0}\frac{1}{x^2}$$

How would you characterize these two? (One of them has a direct bearing on your limit.)

 

[Mark44]

I am not exactly sure about that.

cos(2x)+cos(2a) is NOT an equation, so it can't possibly be an identity.


But that's not a number. I agree that the numerator approaches -1/sqrt(2), which is the same as -sqrt(2)/2. And I agree that the denominator approaches 0.

So this problem is similar to these limits:

$$\lim_{x \to 0}\frac{1}{x}$$
$$\lim_{x \to 0}\frac{1}{x^2}$$

How would you characterize these two? (One of them has a direct bearing on your limit.)

What is it that you're not exactly sure about? If you think that cos(2x)+cos(2a) is an identity, I am absolutely certain that you are wrong.

Are you unsure that your limit is related to one of the ones I gave, you can start by answering my question.