Lim x→7π/4 of (cosx + sinx)/(cos2x)

  • Thread starter Rasine
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In summary, the value of the limit as x approaches 7π/4 for (cosx + sinx)/(cos2x) is undefined due to the denominator approaching 0. To simplify the expression before taking the limit, one can use the trigonometric identity cos2x = 1 - 2sin²x. It is not possible to evaluate the limit numerically as it results in an undefined value. This expression is significant in trigonometry and calculus as it involves trigonometric functions and their limits. The limit can be evaluated at other values of x, but it will only be defined if the denominator is not equal to 0.
  • #1
Rasine
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lim x-> 7pi/4 (cosx+sinx)/(cos2x)

so what i was trying to do here was replace cos2x with the double angle id which is cos^2x-sin^2x

then i factor that and cancle out cosx+sinx

now i have 1/cosx-sinx...so i tried direct sub. and i get 1/sqroot2/2+squroot2/2 which i end up with sqroot 2 but that isn't the right answer

isn't cos 7(pi/4)= squroot 2/2 and sin 7(pi/4)= -squroot2/2?


please show me where i am going wrong
 
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  • #2
You should end up with [itex] \frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2} [/itex].
 
  • #3
He means you did your algebra wrong btw.
 

1. What is the value of the limit as x approaches 7π/4 for (cosx + sinx)/(cos2x)?

The value of the limit is undefined. This is because when x approaches 7π/4, the denominator (cos2x) approaches 0, resulting in an undefined value.

2. How do you simplify the expression (cosx + sinx)/(cos2x) before taking the limit?

To simplify the expression, you can use the trigonometric identity cos2x = 1 - 2sin²x. This will result in the expression (cosx + sinx)/(1 - 2sin²x).

3. Is it possible to evaluate the limit as x approaches 7π/4 numerically?

No, it is not possible to evaluate the limit numerically as it results in an undefined value. This means that the expression does not approach a specific value as x approaches 7π/4.

4. What is the significance of the expression (cosx + sinx)/(cos2x) in mathematics?

This expression is significant in trigonometry and calculus as it involves trigonometric functions and their limits. It also demonstrates the importance of using trigonometric identities to simplify expressions before taking limits.

5. Can the limit of (cosx + sinx)/(cos2x) be evaluated at any other values of x?

Yes, the limit can be evaluated at other values of x, but the value will only be defined if the denominator (cos2x) is not equal to 0. Some examples of values where the limit is defined are x = 0, π/2, and π.

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