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Find the limit: lim (x -> 0) (1 - cos(2x))/x^2
Find the limit: lim (x -> 0) (1 - cos(2x))/x^2
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Q1
Find the limit: lim (x -> 0) (1 - cos(2x))/x^2
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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.
Questions & Step-by-step Solutions
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Q: Find the limit: lim (x -> 0) (1 - cos(2x))/x^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.
Steps: 7
Show Steps
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.
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