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.