# Limits at Infinity for Polynomial Functions

## Understanding Limits Using Cardinal Directions Analogy

In calculus, we often explore how variables behave as they approach infinity (`∞`) or negative infinity (`-∞`). Let's use the cardinal directions (North, South, East, West) as an analogy to understand these concepts better.

### 1. When x Approaches Infinity (`x → ∞`)

- **x → ∞ (East)**: Imagine walking endlessly towards the East. This is like `x` increasing without limits, helping us understand a function's behavior as `x` gets very large.
- **Example**: In `f(x) = 1/x`, as `x` walks East (towards ∞), `f(x)` gradually decreases to 0.

### 2. When x Approaches Negative Infinity (`x → -∞`)

- **x → -∞ (West)**: Think of walking infinitely towards the West. Here, `x` decreases without bounds, representing a function's behavior as `x` extends far in the negative direction.
- **Example**: For `f(x) = x^2`, as `x` walks West (towards -∞), `f(x)` escalates towards ∞.

### 3. When y Approaches Infinity (`y → ∞`)

- **y → ∞ (North)**: This is like moving upwards or Northwards without end. It describes situations where a function's output (`y`) rises indefinitely.
- **Example**: In `f(x) = e^x`, as `x` increases, `y` (or `f(x)`) heads North, approaching ∞.

### 4. When y Approaches Negative Infinity (`y → -∞`)

- **y → -∞ (South)**: Imagine descending endlessly towards the South. This represents a function's output (`y`) falling without limit.
- **Example**: With `f(x) = -e^x`, as `x` goes up, `y` (or `f(x)`) goes South, approaching -∞.

### Combining Directions for x and y Approaching Infinity

1. **x → ∞ and y → ∞ (East and North)**: Both the input (`x`) and output (`y`) of the function become large, like moving Northeast endlessly.
- **Example**: For `f(x) = x^2`, as `x` moves East (towards ∞), `y` also heads North (towards ∞).

2. **x → ∞ and y → -∞ (East and South)**: The input (`x`) heads East, becoming large, but the output (`y`) goes South, becoming large in the negative sense.
- **Example**: For `f(x) = -x^2`, as `x` heads East (towards ∞), `y` heads South (towards -∞).

3. **x → -∞ and y → ∞ (West and North)**: The input (`x`) heads West, becoming large in the negative sense, but the output (`y`) goes North, becoming large.
- **Example**: For `f(x) = -x^3`, as `x` heads West (towards -∞), `y` heads North (towards ∞).

4. **x → -∞ and y → -∞ (West and South)**: Both the input (`x`) and output (`y`) of the function become large in the negative sense, akin to moving Southwest without end.
- **Example**: For `f(x) = x^3`, as `x` moves West (towards -∞), `y` also heads South (towards -∞).

The following graph and legend table summarizes the above.


| Label | Description | Direction Analogy |
|-------|-------------|-------------------|
| **A** | x → -∞,<br> y → ∞ | Northwest (x decreases infinitely, y increases infinitely) |
| **B** | x → finite number,<br> y → ∞ | North (x approaches a specific value, y increases infinitely) |
| **C** | x → ∞, <br>y → ∞ | Northeast (x and y both increase infinitely) |
| **D** | x → ∞,<br> y → finite number | East (x increases infinitely, y approaches a specific value) |
| **E** | x → ∞, <br>y → -∞ | Southeast (x increases infinitely, y decreases infinitely) |
| **F** | x → finite number, <br>y → -∞ | South (x approaches a specific value, y decreases infinitely) |
| **G** | x → -∞, <br>y → -∞ | Southwest (x and y both decrease infinitely) |
| **H** | x → -∞,<br> y → finite number | West (x decreases infinitely, y approaches a specific value) |
---

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### Basic Examples

#### Example 1


| Function | Degree | Leading Coefficient | End Behavior <br> (x → ∞) | End Behavior <br> (x → -∞) |
|----------|--------|---------------------|----------------------|-----------------------|
| $y = x^2$ | Even | Positive | y → ∞ | y → ∞ |

#### Example 2


| Function | Degree | Leading Coefficient | End Behavior <br> (x → ∞) | End Behavior <br> (x → -∞) |
|----------|--------|---------------------|----------------------|-----------------------|
| $y = -x^2$ | Even | Negative | y → -∞ | y → -∞ |

#### Example 3


| Function | Degree | Leading Coefficient | End Behavior <br> (x → ∞) | End Behavior <br> (x → -∞) |
|----------|--------|---------------------|----------------------|-----------------------|
| $y = x^3$ | Odd | Positive | y → ∞ | y → -∞ |

#### Example 4


| Function | Degree | Leading Coefficient | End Behavior <br> (x → ∞) | End Behavior <br> (x → -∞) |
|----------|--------|---------------------|----------------------|-----------------------|
| $y = -x^3$ | Odd | Negative | y → -∞ | y → ∞ |
---
### Polynomial End Behavior Summary Table
Understanding how polynomial functions behave as $x$ approaches infinity or negative infinity is crucial. This behavior depends on the polynomial's degree and its leading coefficient. Below is a table summarizing these relationships:

| Degree | Leading Coefficient | $\displaystyle \lim_{x \to -\infty} f(x)$ | $\displaystyle \lim_{x \to \infty} f(x)$ |
|--------|---------------------|-----------------------------|-----------------------------|
| Odd | Positive | $-\infty$ | $\infty$ |
| Odd | Negative | $\infty$ | $-\infty$ |
| Even | Positive | $-\infty$ | $-\infty$ |
| Even | Negative | $\infty$ | $\infty$ |
| Zero | Any | Constant Value | Constant Value |

## Key Points

- **Leading Coefficient**: This is the coefficient of the term with the highest power in the polynomial.
- **Degree of the Polynomial**: This is the highest power of $x$ in the polynomial.
- **Odd Degree**: Polynomials with an odd degree will have opposite limits at $\pm\infty$.
- **Even Degree**: Polynomials with an even degree will have the same limits at $\pm\infty$, determined by the sign of the leading coefficient.
- **Zero Degree**: Constant functions have the same limit at $\pm\infty$, equal to the constant value.

## Example

Consider a polynomial function $f(x) = ax^n + \dots$, where $a$ is the leading coefficient and $n$ is the degree. The behavior of $f(x)$ as $x$ approaches infinity or negative infinity can be determined using the above table.

## Conclusion

The limit at infinity for polynomial functions is significantly influenced by the polynomial's leading coefficient and degree. This table serves as a guide for predicting the end behavior of such functions.

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## Why the Leading Term of a Polynomial Determines End Behavior.

## Example: Demonstrating End Behavior with $y = x^3 + x$

Consider the polynomial $y = x^3 + x$. Let's see how the leading term ($x^3$) dominates the behavior when `x` is a large number like `1000`.

### Polynomial: y = x^3 + x

- **Leading Term:** `x^3`
- **Other Term:** `x`

### Plugging in x = 1000

1. **Calculate y using the full polynomial:**
- `y = (1000)^3 + 1000`
- `y = 1,000,000,000 + 1000`
- `y ≈ 1,000,000,000` (1 billion)

2. **Calculate y using only the leading term:**
- `y ≈ (1000)^3`
- `y ≈ 1,000,000,000` (1 billion)

### Observation

- When `x = 1000`, the value of `y` using the full polynomial (`x^3 + x`) is almost the same as the value using only the leading term (`x^3`).
- The contribution of the `x` term (1000) is negligible compared to the contribution of the `x^3` term (1 billion) when `x` is large.

### Conclusion

- This example demonstrates that for large values of `x`, the leading term (`x^3` in this case) overwhelmingly determines the behavior of the polynomial.
- As `x` becomes very large, the impact of other, lower-degree terms (like the `x` term here) becomes insignificant in comparison.

# Examples

Polynomial functions show predictable behavior as $x$ approaches infinity or negative infinity. Here are detailed examples explaining how these limits work for various types of polynomials:

## Example 1: Linear Polynomial

### Function: $f(x) = 2x + 3$
- **Behavior**: In linear polynomials, the highest-degree term (here, $2x$) dictates the behavior as $x$ goes to infinity.
- **As $x \to \infty$**: The value of $2x$ becomes very large, leading $f(x)$ to increase indefinitely.
- **Limit**: $\displaystyle \lim_{x \to \infty} f(x) = \infty$.
- **As $x \to -\infty$**: Similarly, when approaching negative infinity, $2x$ becomes increasingly negative, driving $f(x)$ to decrease without bound.
- **Limit**: $\displaystyle \lim_{x \to -\infty} f(x) = -\infty$.
- **Conclusion**: The linear term determines that the function heads towards infinity or negative infinity depending on the direction of $x$.

## Example 2: Quadratic Polynomial

### Function: $g(x) = x^2 - 5x + 6$
- **Behavior**: In quadratic polynomials, the squared term ($x^2$) is most influential for large values of $x$.
- **As $x \to ±\infty$**: The $x^2$ term grows faster than the linear term or constant, causing $g(x)$ to increase without bound in both directions.
- **Limit**: $\displaystyle \lim_{x \to \infty} g(x) = \infty$ and $\displaystyle \lim_{x \to -\infty} g(x) = \infty$.
- **Conclusion**: The positive squared term ensures the function heads towards infinity as $x$ grows in either direction.

## Example 3: Cubic Polynomial

### Function: $h(x) = -3x^3 + 4x^2 + 2x$
- **Behavior**: For cubic polynomials, the cubic term ($-3x^3$) is the most significant for large $x$ values.
- **As $x \to \infty$**: The negative coefficient of $x^3$ means the function decreases to negative infinity.
- **Limit**: $\displaystyle \lim_{x \to \infty} h(x) = -\infty$.
- **As $x \to -\infty$**: When approaching negative infinity, the negative cubic term makes the function increase to positive infinity.
- **Limit**: $\displaystyle \lim_{x \to -\infty} h(x) = \infty$.
- **Conclusion**: The sign of the leading term dictates the function's direction towards positive or negative infinity.

## Example 4: Higher-Degree Polynomial

### Function: $k(x) = 4x^4 - 2x^3 + x - 5$
- **Behavior**: In higher-degree polynomials, the term with the highest degree ($4x^4$ here) dominates as $x$ becomes large.
- **As $x \to ±\infty$**: The positive quartic term ensures that the function's value grows without bound, regardless of $x$'s direction.
- **Limit**: $\lim_{x \to \infty} k(x) = \infty$ and $\lim_{x \to -\infty} k(x) = \infty$.
- **Conclusion**: The highest-degree term, especially its sign and magnitude, determines the function's limit at infinity.

## Conclusion

The limit of a polynomial function at infinity is predominantly influenced by its highest-degree term. The magnitude and sign of this term dictate whether the function approaches positive or negative infinity or stays bounded as $x$ increases or decreases indefinitely.