Use l’Hôpital’s rule to find the limits in Exercises 7–52.
44. lim (x → 0⁺) (csc x - cot x + cos x)

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First, rewrite the expression to understand its behavior as \(x \to 0^+\). The limit is \(\lim_{x \to 0^+} (\csc x - \cot x + \cos x)\).

Recall the definitions: \(\csc x = \frac{1}{\sin x}\) and \(\cot x = \frac{\cos x}{\sin x}\). Substitute these into the expression to get \(\frac{1}{\sin x} - \frac{\cos x}{\sin x} + \cos x\).

Combine the terms with common denominator \(\sin x\): \(\frac{1 - \cos x}{\sin x} + \cos x\).

Analyze the limit of \(\frac{1 - \cos x}{\sin x}\) as \(x \to 0^+\). Both numerator and denominator approach 0, so this is an indeterminate form \(\frac{0}{0}\), which allows the use of l’Hôpital’s Rule.

Apply l’Hôpital’s Rule by differentiating numerator and denominator separately: differentiate \(1 - \cos x\) to get \(\sin x\), and differentiate \(\sin x\) to get \(\cos x\). Then rewrite the limit as \(\lim_{x \to 0^+} \frac{\sin x}{\cos x} + \cos x\) and evaluate the limit.

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주요 개념

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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 Limits and Functions

Understanding the behavior of trigonometric functions such as csc(x), cot(x), and cos(x) near specific points is essential. Knowing their limits and series expansions near zero helps simplify expressions and identify indeterminate forms.

Limit Evaluation Techniques

Besides l’Hôpital’s Rule, techniques like algebraic manipulation, recognizing standard limits, and using trigonometric identities are important. These methods help transform complex expressions into forms suitable for applying l’Hôpital’s Rule or direct substitution.

Use l’Hôpital’s rule to find the limits in Exercises 7–52.44. lim (x → 0⁺) (csc x - cot x + cos x)