Calculate the limit: lim (x -> 0) (x^3)/(sin(x)) (2023)

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Question: Calculate the limit: lim (x -> 0) (x^3)/(sin(x)) (2023)

Options:

Correct Answer: 0

Exam Year: 2023

Solution:

Using the fact that sin(x) ~ x as x approaches 0, we find that lim (x -> 0) (x^3)/(sin(x)) = 0.

Calculate the limit: lim (x -> 0) (x^3)/(sin(x)) (2023)

Calculate the limit: lim (x -> 0) (x^3)/(sin(x)) (2023)

Step 1: Understand the limit we want to calculate: lim (x -> 0) (x^3)/(sin(x)).
Step 2: Recall that as x approaches 0, sin(x) behaves like x. This means sin(x) is approximately equal to x when x is very small.
Step 3: Substitute sin(x) with x in the limit: lim (x -> 0) (x^3)/(sin(x)) becomes lim (x -> 0) (x^3)/x.
Step 4: Simplify the expression: (x^3)/x = x^2.
Step 5: Now we need to find the limit of x^2 as x approaches 0: lim (x -> 0) x^2.
Step 6: As x gets closer to 0, x^2 also gets closer to 0.
Step 7: Therefore, the limit is 0.

Limit of a Function – Understanding how to evaluate the limit of a function as the variable approaches a specific value.
Behavior of Sine Function – Recognizing that sin(x) behaves like x near zero, which is crucial for simplifying the limit.
L'Hôpital's Rule – Applying L'Hôpital's Rule when encountering indeterminate forms, although not necessary in this case.