Find the limit: lim (x -> 0) (1 - cos(2x))/x^2

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Question: Find the limit: lim (x -> 0) (1 - cos(2x))/x^2

Options:

Correct Answer: 2

Solution:

Using the identity 1 - cos(θ) = 2sin^2(θ/2), we have lim (x -> 0) (1 - cos(2x))/x^2 = lim (x -> 0) (2sin^2(x))/x^2 = 2.

Find the limit: lim (x -> 0) (1 - cos(2x))/x^2

Find the limit: lim (x -> 0) (1 - cos(2x))/x^2

Correct Answer: 2

Step 1: Start with the limit expression: lim (x -> 0) (1 - cos(2x))/x^2.
Step 2: Use the trigonometric identity 1 - cos(θ) = 2sin^2(θ/2). Here, θ = 2x, so we rewrite 1 - cos(2x) as 2sin^2(x).
Step 3: Substitute this identity into the limit: lim (x -> 0) (2sin^2(x))/x^2.
Step 4: Factor out the 2 from the limit: 2 * lim (x -> 0) (sin^2(x))/x^2.
Step 5: Recognize that sin(x)/x approaches 1 as x approaches 0. Therefore, (sin^2(x))/x^2 approaches (1^2) = 1.
Step 6: Now, we can evaluate the limit: 2 * 1 = 2.
Step 7: Conclude that the limit is 2.

Limit Evaluation – The question tests the ability to evaluate limits, particularly using trigonometric identities and L'Hôpital's rule if necessary.
Trigonometric Identities – The use of the identity 1 - cos(θ) = 2sin^2(θ/2) is crucial for simplifying the expression.
Indeterminate Forms – Recognizing that the limit initially presents an indeterminate form (0/0) is key to applying the correct techniques.