Find the limit: lim (x -> 0) (x^2)/(sin(x)) (2023)
Practice Questions
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Find the limit: lim (x -> 0) (x^2)/(sin(x)) (2023)
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Questions & Step-by-Step Solutions
Find the limit: lim (x -> 0) (x^2)/(sin(x)) (2023)
Step 1: Understand the limit we want to find: lim (x -> 0) (x^2)/(sin(x)).
Step 2: Recall that as x gets very close to 0, sin(x) gets very close to x.
Step 3: Replace sin(x) with x in the limit: lim (x -> 0) (x^2)/(sin(x)) becomes lim (x -> 0) (x^2)/(x).
Step 4: Simplify the expression: (x^2)/(x) simplifies to x.
Step 5: Now we need to find lim (x -> 0) x, which is simply 0.
Step 6: 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 L'Hôpital's Rule or Taylor series expansion.
Behavior of Trigonometric Functions – Recognizing that as x approaches 0, sin(x) can be approximated by x, which is crucial for simplifying the limit.
