In mathematics, limits describe the value that a function approaches as the input (or independent variable) approaches some value. Limits are fundamental in calculus and analysis, serving as the foundation for defining derivatives, integrals, and continuity.

Key Concepts of Limits:

1. Limit of a function as x approaches a value (c): $\lim_{{x \to c}} f(x) = L$ This reads as "the limit of $f(x)$ as $x$ approaches $c$ is $L$," meaning as $x$ gets closer and closer to $c$, $f(x)$ gets closer to $L$.

2. One-sided limits:

   • Left-hand limit: The value the function approaches as $x$ approaches $c$ from the left ($x \to c^{-}$). $\lim_{{x \to c^{-}}} f(x) = L$
   • Right-hand limit: The value the function approaches as $x$ approaches $c$ from the right ($x \to c^{+}$). $\lim_{{x \to c^{+}}} f(x) = L$

3. Limit at infinity: $\lim_{{x \to \infty}} f(x) = L$ This means that as $x$ increases without bound, the function $f(x)$ approaches the value $L$.

4. Indeterminate forms: Sometimes limits lead to forms like $\frac{0}{0}$ or $\frac{\infty}{\infty}$. In these cases, techniques like L'Hopital's Rule or algebraic manipulation can help to evaluate the limit.

5. Continuity and limits: A function $f(x)$ is continuous at $x = c$ if: $\lim_{{x \to c}} f(x) = f(c)$ This implies that the limit exists, and the function's value at $x = c$ matches the limit.

Would you like any detailed examples or clarifications?

Here are five related questions:

1. How do you compute the limit of a function as $x$ approaches infinity?
2. What is the difference between one-sided and two-sided limits?
3. Can a limit exist if a function is undefined at a point?
4. How does L'Hopital's Rule help with indeterminate forms?
5. How are limits used to define derivatives?

Tip: When evaluating limits, always check for discontinuities or points where the function may be undefined.