Question

Evaluate the limits with either L'Hôpital's rule or previously learned methods. $$ \lim _{x \rightarrow 1} \frac{x-1}{\sin x} $$

Short answer

The limit is 0.

Step-by-step solution

Identify the form of the limit

First, substitute the limit value directly into the function to determine its form: \( \lim_{x \to 1} \frac{x-1}{\sin x} \). Substituting \( x = 1 \), we get \( \frac{1-1}{\sin(1)} = \frac{0}{\sin(1)} \). However, since \( \sin(1) eq 0 \), the limit does not evaluate to an indeterminate form. Thus, it does not directly lead to \( 0/0 \).

Confirm the correct approach

Re-evaluate the initial substitution, considering common misconceptions. Often, one may mistake it as approaching \( 0/0 \), but the correct substitution shows it's merely \( 0 \) divided by a number, leading to \( 0 \). Hence, no complex L'Hôpital's Rule is required.

Simplify further, if necessary

For completeness and understanding, see if any simplification or expansion would clarify the outcome. Here, it wasn't the typical indeterminate form, needing only the knowledge of division by a non-zero constant to evaluate.

Key concepts

L'Hôpital's Rule

L'Hôpital's rule is a useful technique in calculus for finding limits. It comes in handy when we deal with indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). To apply this rule, we take the derivatives of the numerator and the denominator separately, then evaluate their limit instead of the original fraction.
This method simplifies complex expressions that would otherwise be difficult or even impossible to solve directly.

• Example: If we had \( \frac{0}{0} \), we'd apply L'Hôpital's rule by differentiating the top and bottom functions.
• Usage: The rule strictly applies only to certain forms, so always check your limit initially before using it.
• Note: Ensure that the new limit after differentiation is determinate or can be resolved using further calculus techniques.

Understanding this tool is crucial as it will save you time and effort when handling tricky limits.

Indeterminate Forms

In calculus, an indeterminate form occurs when an expression's limit doesn't immediately yield a clear result. The most common indeterminate forms are \( \frac{0}{0} \) and \( \frac{\infty}{\infty} \), but there are others like \( 0 \cdot \infty \), \( \infty - \infty \), and so forth.

These forms require additional analysis because they prevent us from evaluating the limit directly. Often, techniques like algebraic manipulation, L'Hôpital's rule, or series expansion are needed.

• Careful Checking: Always double-check the indeterminate form to confirm its applicability.
• Techniques: Different forms require different techniques, so be prepared to use more than one method.
• Prevention: Sometimes simply rearranging the terms can prevent the form from arising.

Identifying and handling indeterminate forms is a key skill in calculus to reach accurate limit evaluations.

Limit Evaluation Methods

Evaluating limits is a fundamental part of calculus. There are various methods to tackle the challenge, and choosing the correct one is paramount.

Often, direct substitution is the first step. If this results in a clear numerical value, the limit is solved. If not, especially if it produces an indeterminate form, then we explore other methods such as factorization, rationalization, or L'Hôpital's rule.

• Direct Substitution: Simply plug in the approaching value into the expression.
• Algebraic Manipulation: This includes simplifying complex expressions before reevaluating the limit.
• Graphical Approach: Sometimes plotting the function can give us a clear visual understanding of the behavior near the point.

Using variety of methods ensures a comprehensive understanding of how limits behave, leading to accurate evaluations in diverse situations.