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

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

Options:

Correct Answer: -1/2

Solution:

Using the Taylor series expansion for cos(x), we find that lim (x -> 0) (cos(x) - 1)/x^2 = -1/2.

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

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

Correct Answer: -1/2

Step 1: Understand the limit we want to find: lim (x -> 0) (cos(x) - 1)/x^2.
Step 2: Recall the Taylor series expansion for cos(x) around x = 0: cos(x) = 1 - x^2/2 + x^4/24 - ...
Step 3: Substitute the Taylor series into the limit: cos(x) - 1 = -x^2/2 + higher order terms.
Step 4: Rewrite the limit using this substitution: lim (x -> 0) (-(x^2/2) + higher order terms)/x^2.
Step 5: Simplify the expression: lim (x -> 0) (-1/2 + higher order terms/x^2).
Step 6: As x approaches 0, the higher order terms/x^2 approach 0, so we are left with -1/2.
Step 7: Conclude that the limit is -1/2.

Limit Evaluation – Understanding how to evaluate limits, particularly using Taylor series expansions.
Taylor Series – Applying the Taylor series expansion for trigonometric functions to simplify expressions near a point.
Indeterminate Forms – Recognizing and resolving indeterminate forms that arise in limit problems.