What is the limit: lim (x -> 0) (1 - cos(x))/(x^2)? (2022)

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Question: What is the limit: lim (x -> 0) (1 - cos(x))/(x^2)? (2022)

Options:

Correct Answer: 1/2

Exam Year: 2022

Solution:

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

What is the limit: lim (x -> 0) (1 - cos(x))/(x^2)? (2022)

What is the limit: lim (x -> 0) (1 - cos(x))/(x^2)? (2022)

Step 1: Start with the limit we want to find: lim (x -> 0) (1 - cos(x))/(x^2).
Step 2: Use the trigonometric identity: 1 - cos(x) = 2sin^2(x/2).
Step 3: Substitute this identity into the limit: lim (x -> 0) (2sin^2(x/2))/(x^2).
Step 4: Rewrite the limit: lim (x -> 0) (2sin^2(x/2))/(x^2) = 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, we divide by 4 from our earlier step, so we have 8/4 = 2.
Step 10: Thus, the limit is 1/2.

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