1. The problem asks: What is a limit in mathematics? 2. A limit describes the value that a function approaches as the input (or variable) approaches some point. 3. The formal definition: For a function $f(x)$, the limit as $x$ approaches $a$ is $L$ if for every number $\epsilon > 0$, there exists a number $\delta > 0$ such that whenever $0 < |x - a| < \delta$, it follows that $|f(x) - L| < \epsilon$. 4. In simpler terms, as $x$ gets closer and closer to $a$, $f(x)$ gets closer and closer to $L$. 5. Limits help us understand behavior of functions at points where they might not be explicitly defined or where direct substitution is difficult. 6. For example, the limit of $f(x) = \frac{x^2 - 1}{x - 1}$ as $x$ approaches 1 is found by simplifying: $$\frac{x^2 - 1}{x - 1} = \frac{(x-1)(x+1)}{x-1} = x + 1$$ 7. So, as $x \to 1$, $f(x) \to 1 + 1 = 2$. 8. Therefore, the limit is 2. This is the basic concept of limits in calculus and algebra.