Evaluate the limit: lim (x -> 0) (x - sin(x))/x^3 (2022)

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Question: Evaluate the limit: lim (x -> 0) (x - sin(x))/x^3 (2022)

Options:

Correct Answer: 1/6

Exam Year: 2022

Solution:

Using the Taylor series expansion for sin(x), we find that lim (x -> 0) (x - sin(x))/x^3 = 1/6.

Evaluate the limit: lim (x -> 0) (x - sin(x))/x^3 (2022)

Evaluate the limit: lim (x -> 0) (x - sin(x))/x^3 (2022)

Step 1: Understand the limit we want to evaluate: lim (x -> 0) (x - sin(x))/x^3.
Step 2: Recall the Taylor series expansion for sin(x) around x = 0: sin(x) = x - x^3/6 + x^5/120 - ...
Step 3: Substitute the Taylor series into the expression: x - sin(x) = x - (x - x^3/6 + x^5/120 - ...) = x^3/6 - x^5/120 + ...
Step 4: Simplify the expression: (x - sin(x)) = x^3/6 + higher order terms.
Step 5: Now, substitute this back into the limit: lim (x -> 0) (x^3/6 + higher order terms)/x^3.
Step 6: This simplifies to lim (x -> 0) (1/6 + higher order terms/x^3).
Step 7: As x approaches 0, the higher order terms/x^3 approach 0, so we are left with 1/6.
Step 8: Therefore, the limit is 1/6.

Limit Evaluation – Understanding how to evaluate limits, particularly using Taylor series expansions.
Taylor Series – Applying the Taylor series expansion for the sine function to simplify the limit expression.
L'Hôpital's Rule – Recognizing when to apply L'Hôpital's Rule for indeterminate forms, although not used in this solution.