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

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

Options:

Correct Answer: 0

Exam Year: 2022

Solution:

As x approaches 0, x^3 approaches 0 and sin(x) approaches 0, thus the limit is 0.

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

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

Step 1: Identify the limit we need to evaluate: lim (x -> 0) (x^3)/(sin(x)).
Step 2: Understand that as x approaches 0, both the numerator (x^3) and the denominator (sin(x)) approach 0.
Step 3: Recognize that this creates a 0/0 indeterminate form, which means we need to analyze it further.
Step 4: Recall that sin(x) is approximately equal to x when x is very close to 0. This means we can replace sin(x) with x for small values of x.
Step 5: Rewrite the limit using this approximation: lim (x -> 0) (x^3)/(sin(x)) ≈ lim (x -> 0) (x^3)/x.
Step 6: Simplify the expression: (x^3)/x = x^2.
Step 7: Now evaluate the limit of x^2 as x approaches 0: lim (x -> 0) x^2 = 0.
Step 8: Conclude that the limit is 0.

Limit Evaluation – Understanding how to evaluate limits, particularly when both the numerator and denominator approach zero.
L'Hôpital's Rule – Applying L'Hôpital's Rule for indeterminate forms (0/0) to find limits.
Behavior of Functions Near Zero – Recognizing the behavior of polynomial functions and trigonometric functions as they approach zero.