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

lim_x→0 csc 6x sin 7x

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Identify the form of the limit as x approaches 0. The expression csc(6x) is equivalent to 1/sin(6x), so the limit becomes lim_{x→0} (sin(7x) / sin(6x)). This is an indeterminate form 0/0, which suggests the use of l'Hôpital's Rule.

Apply l'Hôpital's Rule, which states that if the limit of f(x)/g(x) as x approaches a value results in an indeterminate form, then lim_{x→c} f(x)/g(x) = lim_{x→c} f'(x)/g'(x), provided the latter limit exists.

Differentiate the numerator and the denominator separately. The derivative of sin(7x) with respect to x is 7cos(7x), and the derivative of sin(6x) with respect to x is 6cos(6x).

Substitute these derivatives back into the limit: lim_{x→0} (7cos(7x) / 6cos(6x)).

Evaluate the new limit as x approaches 0. Since cos(0) = 1, the limit simplifies to 7/6.

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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 determining the continuity and differentiability of functions.

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) results in an indeterminate form, then 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.

Cosecant and Sine Functions

The cosecant function, csc(x), is the reciprocal of the sine function, defined as csc(x) = 1/sin(x). Understanding the behavior of these trigonometric functions, especially near critical points like zero, is essential for evaluating limits involving them. In the given limit, the interaction between csc(6x) and sin(7x) as x approaches zero is key to finding the limit's value.

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.lim_x→0 csc 6x sin 7x