Find the limit: lim (x -> 0) (1 - cos(4x))/(x^2)

Find the limit: lim (x -> 0) (1 - cos(4x))/(x^2)

Step 1: Start with the limit expression: lim (x -> 0) (1 - cos(4x))/(x^2).
Step 2: Use the trigonometric identity 1 - cos(x) = 2sin^2(x/2).
Step 3: Substitute 4x into the identity: 1 - cos(4x) = 2sin^2(4x/2) = 2sin^2(2x).
Step 4: Rewrite the limit using this identity: lim (x -> 0) (2sin^2(2x))/(x^2).
Step 5: Factor out the 2: lim (x -> 0) 2 * (sin^2(2x))/(x^2).
Step 6: Recognize that sin(2x)/(x) approaches 2 as x approaches 0, so (sin(2x)/(x))^2 approaches 4.
Step 7: Therefore, lim (x -> 0) (sin^2(2x))/(x^2) = 4.
Step 8: Multiply by the factor of 2 from Step 5: 2 * 4 = 8.
Step 9: Conclude that the limit is 8.

Limit Evaluation – Understanding how to evaluate limits, particularly using trigonometric identities and L'Hôpital's Rule.
Trigonometric Identities – Applying the identity 1 - cos(x) = 2sin^2(x/2) to simplify expressions involving cosine.
Squeeze Theorem – Recognizing when to apply the Squeeze Theorem for limits involving trigonometric functions.