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

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

Options:

Correct Answer: 1/2

Solution:

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

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

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

Step 1: Start with the limit expression: lim (x -> 0) (1 - cos(x))/(x^2).
Step 2: Use the trigonometric identity: 1 - cos(x) = 2sin^2(x/2).
Step 3: Substitute the identity into the limit: lim (x -> 0) (2sin^2(x/2))/(x^2).
Step 4: Rewrite the limit as: lim (x -> 0) (2sin^2(x/2))/(x/2)^2 * (1/4).
Step 5: Notice that (x/2)^2 = x^2/4, so we can rewrite the limit as: lim (x -> 0) (2sin^2(x/2))/(x^2) = lim (x -> 0) (2sin^2(x/2))/(x^2/4) * 4.
Step 6: This simplifies to: lim (x -> 0) (8sin^2(x/2))/(x^2).
Step 7: As x approaches 0, sin(x/2)/(x/2) approaches 1, so (sin^2(x/2))/(x/2)^2 approaches 1.
Step 8: Therefore, lim (x -> 0) (8sin^2(x/2))/(x^2) = 8 * 1 = 8.
Step 9: Finally, since we factored out a 4 earlier, we divide by 4 to get the final limit: 8/4 = 2.

Limit Evaluation – The question tests the ability to evaluate limits, particularly using trigonometric identities and L'Hôpital's rule.
Trigonometric Identities – The use of the identity 1 - cos(x) = 2sin^2(x/2) is crucial for simplifying the expression.
Indeterminate Forms – Recognizing that the limit results in an indeterminate form (0/0) and knowing how to resolve it.