In mathematics, a limit describes the behavior of a function as its input approaches a certain value. Limits are foundational in calculus and help define important concepts like continuity, derivatives, and integrals.

Types of Limits:

1. Finite Limits: As $x$ approaches some value $a$, the function $f(x)$ approaches a particular number $L$. $\lim_{x \to a} f(x) = L$ Example: $\lim_{x \to 2} (3x + 1) = 7$ As $x$ gets closer to 2, $3x + 1$ approaches 7.

2. Infinite Limits: When the function grows without bound as $x$ approaches a value. $\lim_{x \to a} f(x) = \infty \quad \text{or} \quad \lim_{x \to a} f(x) = -\infty$ Example: $\lim_{x \to 0^+} \frac{1}{x} = \infty$ As $x$ approaches 0 from the right, the function grows infinitely large.

3. Limits at Infinity: This occurs when $x$ tends to infinity or negative infinity, and we observe the behavior of $f(x)$ at those points. $\lim_{x \to \infty} f(x) = L$ Example: $\lim_{x \to \infty} \frac{1}{x} = 0$ As $x$ increases, $\frac{1}{x}$ becomes closer to 0.

4. One-sided Limits: These limits are from either the left or the right of a point.

   • Left-hand limit: $\lim_{x \to a^-} f(x)$
   • Right-hand limit: $\lim_{x \to a^+} f(x)$

5. Limits with Indeterminate Forms: Sometimes limits lead to expressions like $\frac{0}{0}$ or $\frac{\infty}{\infty}$. These are solved using techniques like L'Hopital's Rule.

Would you like a specific example or detailed explanation of any of these types of limits?

Related Questions:

1. How do you compute a limit involving infinity?
2. What are some methods for solving indeterminate forms in limits?
3. How is the concept of limits used to define derivatives?
4. What is L'Hopital's Rule and how does it apply to limits?
5. How do you evaluate one-sided limits?

Tip: Always check if a limit leads to an indeterminate form before deciding the method to solve it.