How do you find the limit of #(1 - cos(2x)) / (x sin(2x)) # as x approaches #0#?
By signing up, you agree to our Terms of Service and Privacy Policy
To find the limit of (1 - cos(2x)) / (x sin(2x)) as x approaches 0, we can use L'Hôpital's Rule. Taking the derivative of the numerator and denominator separately, we get:
Numerator: d/dx (1 - cos(2x)) = 2sin(2x) Denominator: d/dx (x sin(2x)) = sin(2x) + 2x cos(2x)
Now, we can evaluate the limit by substituting x = 0 into the derivatives:
Numerator: 2sin(2(0)) = 0 Denominator: sin(2(0)) + 2(0)cos(2(0)) = 0 + 0 = 0
Since both the numerator and denominator evaluate to 0, we can apply L'Hôpital's Rule again. Taking the derivatives once more:
Numerator: d/dx (2sin(2x)) = 4cos(2x) Denominator: d/dx (sin(2x) + 2x cos(2x)) = 2cos(2x) - 2x sin(2x) + 2cos(2x) - 2x cos(2x) = 4cos(2x) - 2x sin(2x)
Now, substituting x = 0 into the new derivatives:
Numerator: 4cos(2(0)) = 4cos(0) = 4 Denominator: 4cos(2(0)) - 2(0) sin(2(0)) = 4cos(0) = 4
Since both the numerator and denominator evaluate to 4, we can conclude that the limit of (1 - cos(2x)) / (x sin(2x)) as x approaches 0 is 4.
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
- What is the limit of # (1+2x)^(1/x)# as x approaches infinity?
- How do you find the limit of #sec((pix)/6)# as #x->7#?
- How do you find the limit #((1-x)^(1/4)-1)/x# as #x->0#?
- How do you prove that the limit of #((9-4x^2)/(3+2x))=6# as x approaches -1.5 using the epsilon delta proof?
- How do you evaluate #((2x-1)/(2x+5))^(2x+3)# as x approaches infinity?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7