Explain what a limit is using its definition

A limit is a fundamental concept in calculus and mathematical analysis that describes the behavior of a function as its input approaches a certain value. Here's the formal definition:

Let \( f(x) \) be a function defined on an open interval around \( c \) (except possibly at \( c \) itself). The limit of \( f(x) \) as \( x \) approaches \( c \) is \( L \) if, for every number \( \epsilon > 0 \), there exists a number \( \delta > 0 \) such that whenever \( 0 < |x - c| < \delta \), it follows that \( |f(x) - L| < \epsilon \).

In symbols, this is written as:

\[ \lim_{{x \to c}} f(x) = L \]

This means that as \( x \) gets closer and closer to \( c \) (but not equal to \( c \)), the values of \( f(x) \) get closer and closer to \( L \).

### Breaking Down the Definition:

1. **\(\epsilon > 0\)**: This represents how close we want \( f(x) \) to be to \( L \). It's an arbitrary small positive number.
2. **\(\delta > 0\)**: This represents how close \( x \) needs to be to \( c \) to ensure that \( f(x) \) is within \( \epsilon \) of \( L \).
3. **\(|x - c| < \delta\)**: This means \( x \) is within \( \delta \) distance from \( c \) but not equal to \( c \).
4. **\(|f(x) - L| < \epsilon\)**: This means \( f(x) \) is within \( \epsilon \) distance from \( L \).

### Example:

Consider the function \( f(x) = 2x \) and let's find the limit as \( x \) approaches 3.

\[ \lim_{{x \to 3}} 2x = 6 \]

To see why this is true, we can use the formal definition. For every \( \epsilon > 0 \), we need to find a \( \delta > 0 \) such that if \( 0 < |x - 3| < \delta \), then \( |2x - 6| < \epsilon \).

1. Start with \( |2x - 6| < \epsilon \).
2. This simplifies to \( 2|x - 3| < \epsilon \).
3. Dividing both sides by 2 gives \( |x - 3| < \frac{\epsilon}{2} \).

So, we can choose \( \delta = \frac{\epsilon}{2} \). Therefore, for every \( \epsilon > 0 \), we found a \( \delta \) (namely \( \frac{\epsilon}{2} \)) that works, satisfying the definition of the limit.

Would you like more details or have any questions?

Here are some related questions:

1. How do you compute the limit of a polynomial function as \( x \) approaches a point?
2. What is the limit of \( \frac{1}{x} \) as \( x \) approaches infinity?
3. How do you evaluate the limit of a trigonometric function?
4. What does it mean for a limit to be infinite?
5. How does the epsilon-delta definition apply to one-sided limits?
6. Can you explain the concept of a limit at infinity?
7. How do you use limit properties to evaluate complex limits?
8. What are some common techniques for finding limits, such as factoring or rationalizing?

**Tip:** When evaluating limits, it often helps to simplify the function or use limit laws to make the process easier.

Explore the concept of limits in calculus with a detailed explanation of the definition using epsilon-delta. Learn how limits describe the behavior of functions as their inputs approach a specific value. Includes examples and applications. Suitable for college-level students studying calculus.

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