Question

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

Short answer

The limit is 1.

Step-by-step solution

Substitute the limit value

Substitute \(x = 0\) into the expression \(\csc^2 x - \cot^2 x\). We need to see if the result is an indeterminate form. Recall \(\csc x = \frac{1}{\sin x}\) and \(\cot x = \frac{\cos x}{\sin x}\). This substitution gives \(\csc^2(0) - \cot^2(0)\).

Recognize the indeterminate form

Calculate \(\csc^2(0)\) and \(\cot^2(0)\). Since \(\csc(0) = \frac{1}{\sin(0)} = \frac{1}{0}\) and \(\cot(0) = \frac{\cos(0)}{\sin(0)} = \frac{1}{0}\), both are undefined. Hence, the expression \(\csc^2 x - \cot^2 x\) gives \(\frac{1}{0^2} - \left(\frac{1}{0^2}\right) = \frac{0}{0}\), an indeterminate form.

Apply trigonometric identity simplification

Use the identity \(\csc^2 x - \cot^2 x = \frac{1}{\sin^2 x} - \frac{\cos^2 x}{\sin^2 x} = \frac{1 - \cos^2 x}{\sin^2 x}\). Recognize that \(1 - \cos^2 x = \sin^2 x\), hence \(\frac{\sin^2 x}{\sin^2 x} = 1\).

Evaluate the limit

Substitute \(x = 0\) again in the simplified expression. As the simplification gives \(1\), the limit as \(x\) approaches \(0\) of \(\csc^2 x - \cot^2 x\) is \(1\).

Key concepts

Trigonometric Limits

Understanding trigonometric limits is an essential part of calculus, especially when working with trigonometric functions as they approach specific points. When we mention a limit, it basically means finding the value that a function approaches as the input approaches a certain point. In the trigonometric context, we often evaluate limits near standard angles such as 0, \(\frac{\pi}{2}\), \(\pi\), etc.
For trigonometric functions, we use standard limits like \(\lim_{x \to 0} \frac{\sin x}{x} = 1\) to solve more complex problems. Often, these limits involve direct substitution unless they lead to an indeterminate form, in which case more advanced techniques such as l'Hôpital's Rule might be necessary.
In the given problem, the limit \(\lim_{x \to 0}(\csc^2 x - \cot^2 x)\) involves trigonometric functions that approach a zero denominator, hence leading us to an indeterminate form initially.

Indeterminate Forms

Indeterminate forms are expressions that do not have an obvious value, often encountered in limit problems. These typically manifest as \(\frac{0}{0}\), \(\frac{\infty}{\infty}\), \(0 \cdot \infty\), and others.
When substituting \(x = 0\) in \(\csc^2 x - \cot^2 x\), both \(\csc(0)\) and \(\cot(0)\) become problematic due to division by zero, rendering the expression undefined. Hence, it leads to the indeterminate form \(\frac{0}{0}\).
The strategy to tackle these indeterminate forms is to employ algebraic manipulation or utilize l'Hôpital's Rule when applicable. This is crucial for simplifying the expression to reach a determinate form that can be directly calculated.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables. They are particularly valuable in simplifying expressions and solving trigonometric equations.
In the problem at hand, the identity \(\csc^2 x - \cot^2 x = \frac{1 - \cos^2 x}{\sin^2 x}\) is used. Recognizing that \(1 - \cos^2 x\) simplifies to \(\sin^2 x\) due to the Pythagorean identity is key.
This simplification allows us to reduce the expression \(\frac{1 - \cos^2 x}{\sin^2 x}\) directly to 1, making it much easier to evaluate the limit. This step highlights the power of trigonometric identities in resolving complex expressions by transforming them into simpler forms.