60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.

lim_x→0 csc x sin⁻¹ x

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Identify the form of the limit as x approaches 0. The expression csc(x) is 1/sin(x), and sin⁻¹(x) is the inverse sine function. As x approaches 0, both sin(x) and sin⁻¹(x) approach 0, leading to an indeterminate form of 0/0.

Apply l'Hôpital's Rule, which is used to evaluate limits of indeterminate forms like 0/0 or ∞/∞. According to l'Hôpital's Rule, take the derivative of the numerator and the derivative of the denominator separately.

Differentiate the numerator: The derivative of csc(x) is -csc(x)cot(x).

Differentiate the denominator: The derivative of sin⁻¹(x) is 1/√(1-x²).

Re-evaluate the limit using the derivatives: Substitute the derivatives back into the limit expression and evaluate the limit as x approaches 0. Simplify the expression if necessary to find the limit.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They are essential for understanding continuity, derivatives, and integrals. In this context, evaluating the limit as x approaches 0 requires analyzing the behavior of the function near that point.

L'Hôpital's Rule

L'Hôpital's Rule is a method for evaluating limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) leads to an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator. This rule simplifies the process of finding limits in complex expressions.

Cosecant and Inverse Sine Functions

The cosecant function, csc(x), is the reciprocal of the sine function, defined as csc(x) = 1/sin(x). The inverse sine function, sin⁻¹(x), returns the angle whose sine is x. Understanding these functions is crucial for evaluating the limit in the question, as they interact in a way that may lead to an indeterminate form, necessitating the use of L'Hôpital's Rule.

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