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

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

Step 1: Understand the limit we want to find: lim (x -> 0) (x^3)/(sin(x)).
Step 2: Recall that as x approaches 0, sin(x) gets very close to x.
Step 3: This means we can say sin(x) is approximately equal to x when x is near 0.
Step 4: Substitute sin(x) with x in our limit: lim (x -> 0) (x^3)/(sin(x)) becomes lim (x -> 0) (x^3)/x.
Step 5: Simplify (x^3)/x to get x^2.
Step 6: Now we need to find lim (x -> 0) x^2.
Step 7: As x approaches 0, x^2 also approaches 0.
Step 8: 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, particularly using approximations for small values.
Behavior of Sine Function – Recognizing that sin(x) can be approximated by x for values of x close to 0, which is crucial for evaluating limits involving sine.
Indeterminate Forms – Identifying and resolving indeterminate forms that may arise when evaluating limits, particularly 0/0 forms.