A function limit, roughly speaking, describes the behavior of a function around a specific value. Limits play a role in the definition of the derivative and function continuity and are also used in the convergent sequences.

Before getting to the precise definition of a limit, we can investigate limit of a function by plotting it and examining the area around the limit value.

For example, consider the limit (taken from James Stewart's *Calculus Early Transcendentals*, Exercise 11, page 107)

$$ \lim_{x\to 2} \frac{x^2 - x - 6}{x - 2} $$

We can plot this limit using the `matplotlib.pylot`

module. The numpy library is also imported for some convenience functions.

```
import numpy as np
import matplotlib.pyplot as plt
```

```
def f(x):
return (x ** 2 + x - 6) / (x - 2)
```

Get a set of values for $x$ using the `linspace`

function.

```
xvals = np.linspace(0, 2, 100, False)
```

Then plot the function using `matplotlib`

.

```
plt.plot(xvals, f(xvals))
plt.show()
```

The plot shows that as $x$ gets closer to $2$, the function gets close to $5$. One way to verify this is to look closer at the values of the function around the limit.

```
xvals2 = np.linspace(1.90, 2, 100, False)
```

```
f(xvals2)
```

We can also use SymPy's `limit`

function to calculate the limit.

```
from sympy import symbols, limit, sin, cos, init_printing
```

```
x = symbols('x')
init_printing()
```

```
limit((x ** 2 + x - 6) / (x - 2), x, 2)
```

Using this information, we can construct a more precise definition of a limit.

## Definition of a Limit¶

Limits are typically denoted as:

$$ \lim_{x\to a} \space f(x) = L $$

Or, alternatively:

$$ f(x) \rightarrow L, \qquad x \rightarrow a $$

In plain language, we can state the limit as, "the limit of the a function $f(x)$ as $x$ approaches $a$ is equal to $L$. For example, if we were considering the limit:

$$ \lim_{x \to 2} \space f(x) = 5 $$

We can state it as, "the limit of the function $f(x)$ as $x$ approaches $2$ is equal to $5$.

## One-Sided Limits¶

One-sided limits are used to express a limit as it approaches $a$ from a particular direction. The notation is similar to the limit seen above but with a slight change to indicate which direction $x$ is headed.

$$ \lim_{x \to a^+} \space f(x) = L, \qquad \lim_{x \to a^-} \space f(x) = L $$

The notation $x \to a^+$ states we are only interested in values of $x$ that are greater than $a$. Similarly, the notation $x \to a^-$ denotes our desire to investigate values of $x$ less than $a$. These one-sided limits are also referred to as the "right-hand limit" and "left-hand limit", respectively.

For example, consider the function:

$$ \lim_{x \to -3^+} \space \frac{x + 2}{x + 3} $$

The $-3^+$ notation tells us we are only interested values greater than -3; thus the limit is a right-hand limit.

The function is not defined for $x = 3$; therefore we are dealing with an infinite limit. We can see the behavior of the function as $x$ approaches $-3$ by plotting.

```
def f2(x):
return (x + 2) / (x + 3)
```

```
xvals2 = np.linspace(-2, -3, 100, False)
xvals3 = np.linspace(-2.5, -3, 3000, False)
```

```
plt.figure(1, figsize=(14,3))
plt.subplot(121)
plt.plot(xvals2, f2(xvals2))
plt.subplot(122)
plt.plot(xvals3, f2(xvals3))
plt.xlim((-3.005, -2.99))
plt.show()
```

Both graphs approach $x = -3$ from the right and we can see the function quickly drop off as it gets closer to its limit. The graph on the right is a magnified representation of the graph on the left to better illustrate the infinite limit. Therefore, the limit of the function is $-\infty$, which we can confirm with SymPy.

```
limit((x + 2) / (x + 3), x, -3)
```

## References¶

Stewart, J. (2007). Essential Calculus: Early Transcendentals. Belmont, CA: Thomson Higher Education.

Strang, G. (2010). Calculus. Wellesley, MA: Wellesley-Cambridge.