## Understanding Limits in Calculus

### **Introduction to Limits**

#### **What is a Limit?**
- In calculus, a limit is a concept that tells us what value a function (like a formula or equation) is heading towards as its input (usually named 'x') gets closer to a certain point.
- It's written as $\displaystyle \lim_{x \to a} f(x) = L$. This means as `x` gets very close to `a`, the function `f(x)` gets close to the value `L`.

### **Why are Limits Important?**
- Limits help us understand a function's behavior at points where it's not straightforward to evaluate, such as where the function might not be defined or has unusual characteristics.
- They are fundamental in calculus, laying the groundwork for more complex topics like derivatives (how a function changes) and integrals (accumulated totals under a curve).

### **Finding Limits**

#### **1. Using a Table of Values**
- We can estimate a function's limit by choosing inputs (`x` values) close to the point of interest and observing the outputs (`f(x)` values). As `x` gets closer to our point, we look at the trend of `f(x)`.

### Example 1: Finite Limit at a Finite Point

#### Function: $f(x) = \dfrac{x^2 - 1}{x - 1}$

**Objective:** Estimate $\displaystyle \lim_{x \to 1} f(x)$

| x | 0.9 | 0.99 | 0.999 | 1.001 | 1.01 | 1.1 |
|--------|-----|------|-------|-------|------|-----|
| f(x) | 1.9 | 1.99 | 1.999 | 2.001 | 2.01 | 2.1 |

*Observation:* As `x` approaches 1, `f(x)` gets closer to 2. Thus, the estimated limit is 2.

---

### Example 2: Limit at Infinity

#### Function: $f(x) = \dfrac{1}{x}$

**Objective:** Estimate $\displaystyle \lim_{x \to \infty} f(x)$

| x | 10 | 100 | 1000 | 10000 |
|--------|------|-------|-------|-------|
| f(x) | 0.1 | 0.01 | 0.001 | 0.0001|

*Observation:* As `x` becomes very large, `f(x)` gets closer to 0. Therefore, the estimated limit as `x` approaches infinity is 0.

---

### Example 3: Infinite Limit

#### Function: $f(x) = \dfrac{1}{x - 2}$

**Objective:** Estimate $\displaystyle \lim_{x \to 2^+} f(x)$

| x | 2.1 | 2.01 | 2.001 | 2.0001 |
|--------|------|------|-------|--------|
| f(x) | 10 | 100 | 1000 | 10000 |

*Observation:* As `x` approaches 2 from the right, `f(x)` becomes very large. Thus, the estimated limit is infinity.

### Example 4: Limit of a Piecewise Function

#### Function:

$f(x)=\begin{cases} 2x+1 & \text{if } x<3, \\ 5 & \text{if } x=3, \\ x+2 & \text{if } x>3.\end{cases}$

**Objective:** Estimate $\displaystyle \lim_{x \to 3} f(x)$

| x | 2.9 | 2.99 | 2.999 | 3.001 | 3.01 | 3.1 |
|--------|-----|------|-------|-------|------|-----|
| f(x) | 6.8 | 6.98 | 6.998 | 5.001 | 5.01 | 5.1 |

*Observation:*
As $x$ approaches 3 from the left, $f(x)$ approaches 7.

As $x$ approaches 3 from the right, $f(x)$ approaches 5.

Since the limits from the left and right are different, the limit as x approaches 3 does not exist.

### Finding Limits from Graphs

#### **2. Using Graphs**
- By drawing or examining the graph of a function, we can visually inspect the behavior of `f(x)` as `x` approaches a certain value. The graph shows us where the function is heading.

Example :

Here is a graph of a function $f(x)$. The legend is also displayed below.


### Table Legend for the Graph above.

| Symbol | Description | Mathematical Representation |
|--------|-------------------------------------------------------|----------------------------------------|
| A | Limit of $f(x)$ as $x$ approaches $-\infty$ | $\displaystyle \lim_{x \to -\infty} f(x) = -\infty$ |
| B | Limit of $f(x)$ as $x$ approaches $-2$ from the left | $\displaystyle \lim_{x \to -2^-} f(x) = 3$ |
| C | Limit of $f(x)$ as $x$ approaches $-2$ from the right | $\displaystyle \lim_{x \to -2^+} f(x) = 1$ |
| D | Function's value at $x = -2$ | $f(-2) = -2$ |
| E | Limit of $f(x)$ as $x$ approaches $3$ from both sides | $\displaystyle \lim_{x \to 3} f(x) = 4$ |
| F | Function's value at $x = 3$ | $f(3) = 2$ |
| G | Limit of $f(x)$ as $x$ approaches $\infty$ | $\displaystyle \lim_{x \to \infty} f(x) = 3$ |

#### **3. Algebraic Techniques**
- For some functions, we can use algebraic methods like factorization (breaking down into simpler parts), rationalization (dealing with square roots or other roots), or L'Hôpital's Rule (for complex 0/0 or ∞/∞ situations) to find the limit.

### Example 1: Factorization

#### Function: $f(x) = \dfrac{x^2 - 4}{x - 2}$

**Objective:** Find $\displaystyle \lim_{x \to 2} f(x)$ using factorization.

**Solution:**

Factorize the numerator:
$f(x) = \dfrac{(x + 2)(x - 2)}{x - 2}$

Cancel the common terms:
$f(x)= x + 2$

Therefore,
$\displaystyle \lim_{x \to 2} f(x) = 2 + 2 = 4$

### Example 2: Rationalization

#### Function: $f(x) = \frac{\sqrt{x} - 2}{x - 4}$

**Objective:** Find $\displaystyle \lim_{x \to 4} f(x)$ using rationalization.

**Solution:**
Rationalize the numerator:
$\begin{align*}
f(x) = \frac{(\sqrt{x} - 2)(\sqrt{x} + 2)}{(x - 4)(\sqrt{x} + 2)}
= \frac{x - 4}{(x - 4)(\sqrt{x} + 2)}
= \frac{1}{\sqrt{x} + 2}
\end{align*}$

Therefore,
$\displaystyle \lim_{x \to 4} f(x) = \frac{1}{\sqrt{4} + 2} = \frac{1}{4}$

### Example 3: L'Hôpital's Rule

#### Function: $f(x) = \frac{e^x - 1}{x}$

**Objective:** Find $\displaystyle \lim_{x \to 0} f(x)$ using L'Hôpital's Rule, and the derivative rules $\dfrac{d}{dx}(x^n)=nx^{n-1}$ and $\dfrac{d}{dx}(e^x)=e^x$.

**Solution:**
Apply L'Hôpital's Rule (since the limit is of the form 0/0):
Differentiate the numerator and denominator separately:
$T'(x) = \frac{d}{dx}(e^x - 1) = e^x$
$B'(x) = \frac{d}{dx}(x) = 1$

Therefore, using L'Hôpital's Rule:
$\lim_{x \to 0} \frac{e^x - 1}{x} = \lim_{x \to 0} \frac{e^x}{1} = e^0 = 1$

### **Special Types of Limits**

#### **Limits at Infinity**
- These limits describe the behavior of a function as `x` gets extremely large (heads towards infinity). They tell us where the function is 'settling' in the long run.

#### **Infinite Limits**
- An infinite limit occurs when a function grows without bound (becomes infinitely large or small) as `x` approaches a specific value. These are often linked with sharp spikes or drops in the graph.

### **Asymptotes**

#### **Horizontal Asymptotes**
- A horizontal asymptote is a horizontal line that the graph of a function approaches as `x` moves towards infinity or negative infinity. The function gets closer and closer to this line but doesn't touch it.

#### **Vertical Asymptotes**
- A vertical asymptote occurs where a function increases or decreases without limit (becomes infinite) as `x` approaches a specific value. It indicates points where the function is undefined.

Table of Values Example:
## Identifying Asymptotes from a Function's Table of Values

### Horizontal Asymptotes

To find horizontal asymptotes, we look at the behavior of the function's values as $x$ approaches infinity or negative infinity.

| $x$ | $f(x)$ when $x$ is large negative | $f(x)$ when $x$ is large positive |
|----------|------------------------------------|------------------------------------|
| $-10^3$ | -2.001 | |
| $-10^4$ | -2.0001 | |
| $-10^5$ | -2.00001 | |
| $10^3$ | | 3.001 |
| $10^4$ | | 3.0001 |
| $10^5$ | | 3.00001 |

*Observation:* As $x$ becomes very large in the negative direction, $f(x)$ approaches -2. As $x$ becomes very large in the positive direction, $f(x)$ approaches 3.

In Limit Notation this is:

$\displaystyle \lim_{x \to -\infty} f(x)=-2$ and $\displaystyle \lim_{x \to \infty} f(x)=3$.

Thus, we have two horizontal asymptotes at $y = -2$ and $y = 3$.

### Vertical Asymptotes

Vertical asymptotes occur where the function's values increase or decrease without bound as $x$ approaches a specific value from the left or right.

| $x$ approaching value | $f(x)$ from left | $f(x)$ from right |
|--------------------------|-------------------------|-------------------------|
| 0.9 | 50 | |
| 0.99 | 500 | |
| 0.999 | 5000 | |
| 1.001 | | -5000 |
| 1.01 | | -500 |
| 1.1 | | -50 |

*Observation:* As $x$ approaches 1 from the left, $f(x)$ increases without bound, indicating a vertical asymptote at $x = 1$. As $x$ approaches 1 from the right, $f(x)$ decreases without bound, confirming the vertical asymptote at the same point.

In Limit Notation this is:

$\displaystyle \lim_{x \to 1^-} f(x)=\infty$ and $\displaystyle \lim_{x \to 1^+} f(x)=-\infty$.

## Graph Example
Here is a graph of $f(x)=\dfrac{5(x+1)(x-3)}{(x+4)(x-2)}$


### Table Legend for Graph

| Symbol | Description | Mathematical Representation | Limit Notation |
|--------|-----------------------------------------------------------------|----------------------------------------|-------------------------------------------------|
| A | Limit of $f(x)$ as $x$ approaches $-\infty$ | | $\displaystyle \lim_{x \to -\infty} f(x) = 5$ |
| B | Limit of $f(x)$ as $x$ approaches $-4$ from the left | | $\displaystyle \lim_{x \to -4^-} f(x) = \infty$ |
| C | Limit of $f(x)$ as $x$ approaches $-4$ from the right | | $\displaystyle \lim_{x \to -4^+} f(x)=-\infty$ |
| D | Limit of $f(x)$ as $x$ approaches $2$ from the left | | $\displaystyle \lim_{x \to 2^-} f(x)=\infty$ |
| E | Limit of $f(x)$ as $x$ approaches $2$ from the right | | $\displaystyle \lim_{x \to 2^+} f(x)=-\infty$ |
| F | Limit of $f(x)$ as $x$ approaches $\infty$ | | $\displaystyle \lim_{x \to \infty} f(x) = 5$ |
| G | Horizontal asymptote | $y=5$ |
| H | Vertical Asymptote at $x = -4$ | $x = -4$ | |
| J | Vertical Asymptote at $x = 2$ | $x = 2$ | |

### **Conclusion**
- Learning about limits is crucial in understanding how functions behave, and it's the first step in exploring the fascinating world of calculus.

### **Additional Resources**
- For further learning, consider introductory calculus textbooks or online educational resources that provide more examples and detailed explanations.