In mathematics, "lim" stands for the limit of a function, which describes the behavior of a function as the input approaches a particular value. Limits are fundamental in calculus and analysis and help define key concepts like continuity, derivatives, and integrals.

Example:

For a function $f(x)$, the limit as $x$ approaches some value $a$ is written as:

$\lim_{{x \to a}} f(x)$

This expression means "the value that $f(x)$ approaches as $x$ gets closer to $a$." The limit might exist, not exist, or tend to infinity.

Types of Limits:

1. Finite Limits: When the function approaches a finite value as $x$ approaches $a$.
2. Infinite Limits: When the function grows without bound as $x$ approaches $a$.
3. One-sided Limits: Limit as $x$ approaches $a$ from the left ($\lim_{{x \to a^-}} f(x)$) or the right ($\lim_{{x \to a^+}} f(x)$).
4. Limits at Infinity: As $x$ approaches $\infty$ or $-\infty$.

Example Problem:

Find the limit: $\lim_{{x \to 2}} (x^2 - 4)$

Solution: Substitute $x = 2$: $\lim_{{x \to 2}} (x^2 - 4) = 2^2 - 4 = 0$

Would you like further details or explanations on limits?

5 Related Questions:

1. How do you compute a limit when substitution doesn't work (e.g., indeterminate forms)?
2. What are one-sided limits and how are they different from regular limits?
3. How do limits relate to the concept of continuity?
4. What are L'Hopital's Rule and its application in limits?
5. How do limits help define derivatives in calculus?

Tip: When dealing with limits, always check for the possibility of indeterminate forms like $\frac{0}{0}$ or $\frac{\infty}{\infty}$, as these often require special techniques like factoring or L'Hopital's Rule.