Problem 30 - Thomas Calculus

Question

L'Hopital's Rule does not help with the limits in Exercises \(27-30 .\) Try it; you just keep on cycling. Find the limits some other way. $$ \lim _{x \rightarrow 0^{+}} \frac{\cot x}{\csc x} $$

Step-by-Step Solution

Answer

The limit is 0 as \( x \to 0^{+} \).

Step 1: Understand the Limit Notation

We want to find the limit of the function \( \frac{\cot x}{\csc x} \) as \( x \) approaches \( 0^{+} \). This means we're considering values of \( x \) that get closer and closer to 0 from the positive side.

Step 2: Express Trigonometric Functions in Terms of Sine and Cosine

Recall that \( \cot x = \frac{\cos x}{\sin x} \) and \( \csc x = \frac{1}{\sin x} \). Substituting these into the limit, we get:\[ \lim_{x \rightarrow 0^{+}} \frac{\frac{\cos x}{\sin x}}{\frac{1}{\sin x}} \]

Step 3: Simplify the Fraction

The expression can be simplified by multiplying the numerator by the reciprocal of the denominator:\[ \frac{\frac{\cos x}{\sin x}}{\frac{1}{\sin x}} = \cos x \times \sin x \]

Step 4: Evaluate the Limit

Now find the limit:\[ \lim_{x \rightarrow 0^{+}} \cos x \times \sin x \]As \( x \to 0^{+} \), we know \( \cos x \rightarrow 1 \) and \( \sin x \rightarrow 0 \). Therefore, \( \cos x \times \sin x \rightarrow 1 \times 0 = 0 \).

Step 5: Conclude the Limit

The limit of the given expression as \( x \to 0^{+} \) is 0. L'Hopital's Rule was not needed because we simplified the expression directly.

Key Concepts

Limit Evaluation

Evaluating limits is a fundamental concept in calculus, helping us understand the behavior of functions as they approach a particular point. In limit evaluation, especially when the function forms an indeterminate expression like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), L'Hôpital's Rule might be the first approach to consider. However, this rule isn't always needed and sometimes doesn't help. Instead, simplifying the expression often reveals a clear path to the solution.

For example, in the problem given, we are trying to find \( \lim_{x \rightarrow 0^{+}} \frac{\cot x}{\csc x} \). As you start by translating the trigonometric functions into their sine and cosine forms, you simplify the expression to something manageable without invoking L'Hôpital's Rule—a method that can sometimes lead to cyclic calculations with no resolution when trigonometric terms are involved.

The key takeaway is that being adept at recognizing when L'Hôpital's Rule might not work effectively saves time and effort, allowing you to focus on direct simplification techniques for smooth and straightforward limit evaluation.

Trigonometric Limits

Trigonometric limits often involve expressions where direct substitution of limits results in an indeterminate form. These expressions can initially appear complex, especially when they involve multiple trigonometric functions. In such cases, rewriting trigonometric functions in terms of sine and cosine can simplify the process significantly.

For the given exercise, \( \frac{\cot x}{\csc x} \) involves both cotangent and cosecant, which can be challenging to work with directly in limits. Remember that:

• \( \cot x = \frac{\cos x}{\sin x} \)
• \( \csc x = \frac{1}{\sin x} \)

By expressing both the numerator and the denominator in terms of sine and cosine, the expression simplifies, ultimately allowing you to find the limit with ease. As \( x \) approaches \( 0^{+} \), despite the small input values, the limits of sine and cosine are well-defined, leading to a more straightforward calculation.

Simplifying Trigonometric Expressions

One of the most valuable tools in evaluating limits involving trigonometric functions is the ability to simplify expressions effectively. Simplification reduces complexity and aids in identifying clear pathways to solutions. Breaking down complex fractions into more intuitive terms using fundamental trigonometric identities is crucial.

Consider the given problem \( \frac{\cot x}{\csc x} \). By converting into sine and cosine, this expression transitions from a more complex fraction into a product form: \( \cos x \times \sin x \). This transformation occurs because dividing by \( \frac{1}{\sin x} \) is the same as multiplying by \( \sin x \). This simple switch often uncovers hidden limits that weren't obvious before.

Such simplifications are vital for avoiding unnecessary calculations with more complicated methods like L'Hôpital's Rule. They also highlight the interconnectedness of trigonometric identities and the importance of mastering these basics to navigate and solve calculus problems efficiently.