Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
93. lim(x→0) (csc(x) - cot(x))

Verified step by step guidance

1. First, recognize that as \(x \to 0\), both \(\csc(x)\) and \(\cot(x)\) approach forms that may lead to an indeterminate expression. Specifically, \(\csc(x) = \frac{1}{\sin(x)}\) and \(\cot(x) = \frac{\cos(x)}{\sin(x)}\), so the expression \(\csc(x) - \cot(x)\) can be rewritten as \(\frac{1}{\sin(x)} - \frac{\cos(x)}{\sin(x)}\).
2. Combine the terms over a common denominator to simplify the expression: \(\csc(x) - \cot(x) = \frac{1 - \cos(x)}{\sin(x)}\).
3. Evaluate the limit \(\lim_{x \to 0} \frac{1 - \cos(x)}{\sin(x)}\). Direct substitution gives \(\frac{1 - 1}{0} = \frac{0}{0}\), which is an indeterminate form, so l’Hôpital’s Rule applies.
4. Apply l’Hôpital’s Rule by differentiating the numerator and denominator separately with respect to \(x\): differentiate numerator \(1 - \cos(x)\) to get \(\sin(x)\), and differentiate denominator \(\sin(x)\) to get \(\cos(x)\).
5. Rewrite the limit as \(\lim_{x \to 0} \frac{\sin(x)}{\cos(x)}\) and then evaluate this new limit by direct substitution.

Key Concepts

l’Hôpital’s Rule

l’Hôpital’s Rule is a method for evaluating limits that result in indeterminate forms like 0/0 or ∞/∞. It states that the limit of a ratio of functions can be found by taking the limit of the ratio of their derivatives, provided certain conditions are met.

Trigonometric Functions and Their Limits

Understanding the behavior of trigonometric functions such as csc(x) and cot(x) near zero is essential. Knowing standard limits like sin(x) ≈ x near zero helps simplify expressions and identify indeterminate forms.

Derivative of Trigonometric Functions

To apply l’Hôpital’s Rule, one must compute derivatives of csc(x) and cot(x). The derivatives are -csc(x)cot(x) and -csc²(x), respectively, which are crucial for simplifying the limit expression after differentiation.