In mathematics, limits describe the value that a function approaches as the input approaches a certain point. Limits are fundamental to calculus and are used to define both derivatives and integrals.

Basic idea of limits:

• Limit of a function: If $f(x)$ approaches a particular value $L$ as $x$ gets arbitrarily close to a specific point $c$, we say that the limit of $f(x)$ as $x$ approaches $c$ is $L$. This is written as: $\lim_{x \to c} f(x) = L$
• One-sided limits: These are limits where $x$ approaches $c$ from one side, either the left ($x \to c^-$) or the right ($x \to c^+$).
  • Left-hand limit: $\lim_{x \to c^-} f(x)$
  • Right-hand limit: $\lim_{x \to c^+} f(x)$

Key types of limits:

1. Finite limits: When the limit $L$ is a specific finite number.
2. Infinite limits: When $f(x)$ increases or decreases without bound as $x$ approaches $c$, such as: $\lim_{x \to c} f(x) = \infty \quad \text{or} \quad \lim_{x \to c} f(x) = -\infty$
3. Limits at infinity: When $x$ approaches infinity or negative infinity and the function approaches a particular value: $\lim_{x \to \infty} f(x) \quad \text{or} \quad \lim_{x \to -\infty} f(x)$

Techniques for solving limits:

• Direct substitution: Substitute the value of $x$ into the function directly.
• Factoring: Factor expressions to cancel terms and simplify.
• Rationalizing: Use rationalization to simplify functions with square roots.
• L'Hopital's Rule: Apply this rule for indeterminate forms (e.g., $\frac{0}{0}$, $\frac{\infty}{\infty}$) by differentiating the numerator and the denominator.

Examples:

1. $\lim_{x \to 2} (x^2 - 4) = 0$ Here, by directly substituting $x = 2$, the result is $0$.

2. $\lim_{x \to \infty} \frac{1}{x} = 0$ As $x$ increases indefinitely, $\frac{1}{x}$ approaches 0.

Would you like to dive into more examples or techniques for limits? Let me know!

Follow-up questions:

1. How do you handle a limit when direct substitution results in $\frac{0}{0}$?
2. What is the difference between a one-sided and a two-sided limit?
3. Can a limit exist if the left-hand limit and the right-hand limit are different?
4. How does L'Hopital's Rule work for limits involving indeterminate forms?
5. What happens to the limit of a rational function as $x \to \infty$?

Tip:

For continuous functions, limits can often be found just by direct substitution, making the process straightforward!