Question

Find each limit. Be sure you have an indeterminate form before applying l'Hôpital's Rule. $$ \lim _{x \rightarrow 0} \frac{2 \csc ^{2} x}{\cot ^{2} x} $$

Short answer

The limit is 2.

Step-by-step solution

Interpret the Function

The function to evaluate the limit for is \( \frac{2 \csc ^{2} x}{\cot ^{2} x} \). Recall that \( \csc x = \frac{1}{\sin x} \) and \( \cot x = \frac{\cos x}{\sin x} \).

Substitute Trigonometric Identities

Substitute the trigonometric identities into the expression: \( 2 \csc^2 x = \frac{2}{\sin^2 x} \) and \( \cot^2 x = \frac{\cos^2 x}{\sin^2 x} \). This gives us the expression \( \frac{2}{\sin^2 x} \div \frac{\cos^2 x}{\sin^2 x} \).

Simplify the Expression

Simplify the division of fractions: \( \frac{2 \cdot \sin^2 x}{\sin^2 x \cdot \cos^2 x} = \frac{2}{\cos^2 x} \). Thus, the expression becomes \( 2 \sec^2 x \).

Evaluate the Limit

The limit we need to evaluate is now \( \lim _{x \rightarrow 0} 2 \sec ^{2} x \). The secant function, \( \sec x = \frac{1}{\cos x} \), so \( \sec^2 x = \frac{1}{\cos^2 x} \). As \( x \to 0 \), \( \cos x \to 1 \), hence \( \sec^2 x \to 1 \).

Final Calculation

Substitute the value of the limit into the simplified expression: \( 2 \times 1 = 2 \).

Key concepts

l'Hôpital's Rule

When evaluating limits in calculus, one powerful tool at your disposal is l'Hôpital's Rule. This rule is especially useful when you encounter an indeterminate form such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).
This happens when both the numerator and the denominator of a function tend to zero, or both tend to infinity, as \( x \) approaches a certain value. The rule states that if the limit leads to these indeterminate forms, you can take the derivative of the numerator and the denominator separately and then evaluate the limit of this new quotient.
However, remember to always verify that you genuinely have an indeterminate form before applying the rule. This ensures that the function satisfies the conditions needed for l'Hôpital's Rule to be valid.

Trigonometric identities

Trigonometric identities are essential tools in calculus and are often used to simplify expressions that involve trigonometric functions. In this exercise, crucial identities used were those for cosecant (\( \csc x \)) and cotangent (\( \cot x \)).
Here's a quick recap:

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

Knowing these identities allows you to transform complex trigonometric expressions into simpler algebraic fractions.
This is vital for the limit evaluation process as it often leads to clearer insights into the problem.

Limit evaluation

Limit evaluation is a fundamental technique in calculus used to determine the behavior of functions as they approach a particular input value.
In simple terms, you are trying to find out what value a function gets closer to as the input (usually \( x \)) approaches some target value.
This technique is crucial when dealing with functions that are not easily computed directly at a certain point, often due to indeterminacy or discontinuity. To evaluate limits effectively:

• Identify if there's an indeterminate form.
• Rewrite the expression using trigonometric identities or algebraic manipulation if needed.
• Apply techniques like l'Hôpital's Rule, if applicable, to simplify evaluation.
• Substitute the target value into the simplified expression to find the limit.

With these steps, evaluating the limit becomes a structured process that often reveals the hidden behavior of functions around challenging points.

Indeterminate forms

In calculus, indeterminate forms are a set of expressions encountered when evaluating limits, where the limit cannot be immediately determined. Common indeterminate forms include \( \frac{0}{0} \), \( \frac{\infty}{\infty} \), \( 0 \cdot \infty \), and \( \infty - \infty \).
These forms are 'indeterminate' because they do not clearly point to a unique limit value, necessitating further manipulation or techniques (like l'Hôpital's Rule) to resolve them. Understanding the nature of indeterminate forms is crucial as it indicates the paths to take in transforming and simplifying the expressions to reach a clear and definitive limit.
A handy tip is always to look for opportunities to simplify such expressions before applying more complex tools, as sometimes even simple algebra or trigonometric identity can resolve an indeterminate form.