Evaluating limits problems is a fundamental skill in calculus, providing insights into the behavior of functions near specific points. Mastering this concept opens the door to understanding continuity, derivatives, and integrals. This post will help you understand and solve limit problems. We will explore how to approach and solve a specific limit problem. The SEO keyphrase will be the focus.

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Let’s explore the concept of evaluating limits, a fundamental aspect of calculus. We will delve into a specific example, providing a step-by-step solution and related problems to solidify understanding. The focus will be on how to solve the given limit problems, ensuring clarity and ease of comprehension. Understanding limits is crucial for grasping concepts like continuity and derivatives.

Understanding the Concept of Limits

The SEO keyphrase, evaluating limits, is a core concept in calculus, essential for understanding how functions behave near certain points. Limits allow us to analyze function behavior as the input approaches a specific value, without necessarily reaching it. They are the foundation for more advanced topics, such as differentiation and integration.

When evaluating limits, we’re essentially asking what value a function approaches as its input gets arbitrarily close to a certain value. This is a crucial skill in calculus. The limit’s value doesn’t always equal the function’s value at that point, especially in cases of discontinuities or undefined points.

Solving the Limit Problem

Applying the Limit Definition

To solve the limit problem, we use the fundamental limit: ##\lim_{x \to 0} \frac{\sin(x)}{x} = 1##. The given problem, involving ##\frac{\sin(5x)}{x}##, requires some manipulation. We aim to transform the expression to fit the known limit form. This involves adjusting the argument of the sine function.

The key is to make the argument of the sine function match the denominator. We can achieve this by multiplying and dividing by 5. This approach helps us to create a form that closely resembles the standard limit, making it easier to evaluate. The core concept here is algebraic manipulation.

Step-by-Step Solution

Multiply and divide by 5: ##\lim_{x \to 0} \frac{\sin(5x)}{x} = \lim_{x \to 0} 5 \cdot \frac{\sin(5x)}{5x}##. Let ##u = 5x##, then as ##x \to 0##, ##u \to 0##. The expression becomes ##5 \cdot \lim_{u \to 0} \frac{\sin(u)}{u}##. This transformation aligns with the known limit, simplifying the calculation.

Using the fundamental limit, we find ##\lim_{u \to 0} \frac{\sin(u)}{u} = 1##. Therefore, the original limit simplifies to ##5 \cdot 1##. This is a direct application of a standard limit, illustrating how algebraic manipulation is used in evaluating limits problems. This step-by-step process is crucial.

Final Solution

The limit is ##5##. This result is obtained by applying the standard limit and basic algebraic manipulation. The solution underscores the importance of recognizing familiar forms and how to adapt the given expression to fit them. The answer is straightforward.

Therefore, ##\lim_{x \to 0} \frac{\sin(5x)}{x} = 5##. This result showcases the application of limits. This means that as x approaches 0, the function ##\frac{\sin(5x)}{x}## approaches 5. This completes the solution.

Similar Problems and Quick Solutions

Problem 1: Evaluate ##\lim_{x \to 0} \frac{\sin(3x)}{x}##

Solution: ##3##

Problem 2: Evaluate ##\lim_{x \to 0} \frac{\tan(2x)}{x}##

Solution: ##2##

Problem 3: Evaluate ##\lim_{x \to 0} \frac{\sin(x)}{\sin(2x)}##

Solution: ##\frac{1}{2}##

Problem 4: Evaluate ##\lim_{x \to 0} \frac{x}{\sin(4x)}##

Solution: ##\frac{1}{4}##

Problem 5: Evaluate ##\lim_{x \to 0} \frac{\sin(x) – x}{x^3}##

Solution: ##-\frac{1}{6}##

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