91Ó°ÊÓ

In calculus, a limit describes the behavior of a function as the input approaches a particular value. In our exercise, we are evaluating \(\lim _{\theta \rightarrow 0} \frac{\theta+\sin \theta}{\tan \theta}\). Limits are crucial in calculus as they form the foundation of derivatives and integrals. When you have a complex fraction or expression, evaluating a limit helps understand how the overall expression behaves as it approaches a certain point.

Limits can be tricky, especially when dealing with indeterminate forms like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). In such cases, techniques like l'Hôpital's Rule become very handy. It's important to first substitute the number you're approaching and see if it leads to a definitive form or an indeterminate form. In this exercise, substituting gives us \(\frac{0}{0}\), which indicates the need for l'Hôpital's Rule.

Trigonometric functions are functions of angles and are used extensively in various areas of mathematics and physics. These functions include sine, cosine, and tangent, among others. In this exercise, we deal with the sine function \(\sin \theta\) and the tangent function \(\tan \theta\).

- **Sine function (\(\sin \theta\))**: This function gives the y-coordinate of a point on the unit circle as the angle \(\theta\) increases.- **Tangent function (\(\tan \theta\))**: It is defined as the ratio of \(\sin \theta\) to \(\cos \theta\) and can be expressed as \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).
Understanding how these functions behave as \(\theta\) approaches 0 is essential when solving limits involving trigonometric functions. As \(\theta\) approaches 0, both \(\sin \theta\) and \(\tan \theta\) approach 0, leading to special conditions that often require careful evaluation using tools like l'Hôpital's Rule.

Derivatives measure the rate at which a quantity changes. They are the fundamental concept of differential calculus. In this exercise, we need to find the derivatives of both the numerator and the denominator of the fraction.

For the **numerator** \(\theta + \sin\theta\), the derivative is:

• The derivative of \(\theta\) with respect to \(\theta\) is 1.
• The derivative of \(\sin \theta\) is \(\cos \theta\).

Therefore, the derivative of the numerator is \(1 + \cos \theta\).

For the **denominator** \(\tan\theta\), the derivative is \(\sec^2 \theta\), since the derivative of \(\tan\theta\) is \(\sec^2 \theta\).
Obtaining these derivatives is crucial to apply l'Hôpital's Rule. Once the derivatives are found, they replace the original functions in the limit, allowing for a new evaluation that resolves the indeterminate form effectively.

Indeterminate forms occur in calculus when evaluating limits and the limit appears in a form that is undefined, such as \(\frac{0}{0}\). These forms require special techniques or manipulations to evaluate properly.

In the case of this exercise, when we substitute \(\theta = 0\), both the numerator \(\theta + \sin \theta\) and the denominator \(\tan \theta\) evaluate to 0, leading to \(\frac{0}{0}\). This signal tells us that the limit is not straightforward to evaluate.

**l'Hôpital's Rule** is a strategic approach for handling such indeterminate forms. It states that if the limit yields \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), one can take the derivatives of the numerator and denominator separately, then evaluate the limit of this new fraction. In our exercise, applying l'Hôpital's Rule allowed us to change the original limit to a new limit involving \(\frac{1 + \cos \theta}{\sec^2\theta}\), which we could then easily evaluate as \(\theta\) approaches 0, leading to the answer of 2. This method greatly simplifies solving limits that initially seem impossible to resolve.