The Limit of a Function: A Simple and Clear Explanation

The limit of a function describes the behavior of the function $f(x)$ as $x$ approaches a specific value. Limits are foundational in calculus and are used to define derivatives, integrals, and continuity.

---

1. What is a Limit?

If a function $f(x)$ gets arbitrarily close to a single value $L$ as $x$ approaches some value $c$, we say: $\lim_{x \to c} f(x) = L$ This means that as $x$ gets closer and closer to $c$ (but not necessarily equal to $c$), the function $f(x)$ approaches $L$.

---

2. Key Cases

• Finite Value: $\lim_{x \to 2} (3x) = 6$. As $x$ gets close to $2$, $3x$ gets close to $6$.
• Infinite Limit: When $f(x)$ increases or decreases without bound as $x$ approaches $c$, we write $\lim_{x \to c} f(x) = \infty$ or $-\infty$.
• Does Not Exist (DNE): If there is no single value the function approaches, the limit does not exist.

---

3. Left-Hand and Right-Hand Limits

Sometimes, a function behaves differently as $x$ approaches $c$ from the left or the right:

• Left-hand limit: $\lim_{x \to c^-} f(x)$: As $x$ approaches $c$ from values less than $c$.
• Right-hand limit: $\lim_{x \to c^+} f(x)$: As $x$ approaches $c$ from values greater than $c$.

For the limit to exist, both left-hand and right-hand limits must be equal: $\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = \lim_{x \to c} f(x)$

---

4. Important Techniques

To compute limits, we use:

1. Substitution: Directly substitute $x = c$ if the function is continuous.
2. Factoring: Factorize to cancel terms when there are indeterminate forms like $0/0$.
3. Rationalization: Multiply by the conjugate for functions with square roots.
4. L’Hôpital’s Rule: Use derivatives to resolve $0/0$ or $\infty/\infty$.
5. Graph Analysis: Observe how $f(x)$ behaves near $c$ using a graph.

---

5. Special Limits

1. Limits at Infinity:
   • As $x \to \infty$, $\frac{1}{x} \to 0$.
2. Trigonometric Limits:
   • $\lim_{x \to 0} \frac{\sin x}{x} = 1$.
3. Exponential and Logarithmic Limits:
   • $\lim_{x \to \infty} e^{-x} = 0$.

---

Key Notes to Remember

• A limit focuses on what happens near $x = c$, not necessarily at $x = c$.
• Indeterminate forms (e.g., $0/0, \infty - \infty$) require special techniques to evaluate.
• Graphical intuition helps clarify challenging limits.
• Infinite limits describe unbounded behavior, while limits at infinity describe behavior as $x$ grows without bound.

---

Example

Evaluate $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$:

1. Direct substitution gives $\frac{0}{0}$ (indeterminate).
2. Factorize $\frac{(x - 2)(x + 2)}{x - 2}$.
3. Cancel $(x - 2)$: $\lim_{x \to 2} (x + 2) = 4$.

---

Do you want more examples or specific problems to solve?
Here are 5 related questions to practice:

1. What is the meaning of $\lim_{x \to \infty} \frac{1}{x}$?
2. How do you solve $\lim_{x \to 0^+} \ln(x)$?
3. Can you compute $\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$?
4. Why does the limit $\lim_{x \to 0} \frac{\sin(x)}{x}$ equal 1?
5. How does L’Hôpital’s Rule simplify the evaluation of $\lim_{x \to \infty} \frac{x}{e^x}$?

Tip: Always check continuity first; limits of continuous functions are straightforward to compute!