17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.


lim_x→0 x csc x

Verified step by step guidance

1. First, recognize that the limit \( \lim_{x \to 0} x \csc x \) can be rewritten using the identity \( \csc x = \frac{1}{\sin x} \). Thus, the expression becomes \( \lim_{x \to 0} \frac{x}{\sin x} \).
2. Observe that as \( x \to 0 \), both the numerator and the denominator approach 0, creating an indeterminate form \( \frac{0}{0} \). This is a situation where l'Hôpital's Rule can be applied.
3. Apply l'Hôpital's Rule, which states that for limits of the form \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), the limit \( \lim_{x \to a} \frac{f(x)}{g(x)} \) can be evaluated as \( \lim_{x \to a} \frac{f'(x)}{g'(x)} \), provided the limit on the right exists.
4. Differentiate the numerator \( x \) to get \( 1 \), and differentiate the denominator \( \sin x \) to get \( \cos x \). The limit now becomes \( \lim_{x \to 0} \frac{1}{\cos x} \).
5. Evaluate the new limit \( \lim_{x \to 0} \frac{1}{\cos x} \). Since \( \cos 0 = 1 \), the limit simplifies to \( \frac{1}{1} \).

Key Concepts

Limits

Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding the function's behavior near points of interest, including points of discontinuity or where the function is not explicitly defined. Evaluating limits is crucial for defining derivatives and integrals, which are core components of calculus.

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms, such as 0/0 or ∞/∞. The rule 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 separately. This process can be repeated if the result remains indeterminate, making it a powerful tool for limit evaluation.

Cosecant Function (csc)

The cosecant function, denoted as csc(x), is the reciprocal of the sine function, defined as csc(x) = 1/sin(x). It is important in trigonometry and calculus, particularly when dealing with limits involving trigonometric functions. Understanding the behavior of csc(x) as x approaches certain values, such as 0, is essential for evaluating limits that involve this function.