Implementing Limits in Python

Before exploring how limits behave visually, you need to know how to compute them directly using the sympy library. Here are three common types of limits you'll encounter.

1. Finite Limit

This example shows a function that approaches a specific finite value as $x \to 2$.

2. Limit That Does Not Exist

Here, the function behaves differently from the left and right sides, so the limit does not exist.

3. Infinite Limit

This example shows a function that approaches zero as (x) grows infinitely large.

These short snippets demonstrate how to use sympy.limit() to compute different types of limits - finite, undefined, and infinite - before analyzing them graphically

Defining the Functions

f_diff = (2 - x)  # Approaches +∞ as x → -∞ and -∞ as x → +∞
f_same = 1 / x  # Approaches 0 as x → ±∞
f_special = sp.sin(x) / x  # Special limit sin(x)/x

• f_diff: a simple linear function where the left-hand and right-hand limits diverge;
• f_same: the classic reciprocal function, approaching the same limit from both sides;
• f_special: a well-known limit in calculus, which equals 1 as $x \to 0$.

Handling Division by Zero

y_vals_same = [f_same.subs(x, val).evalf() if val != 0 else np.nan for val in x_vals]
y_vals_special = [f_special.subs(x, val).evalf() if val != 0 else 1 for val in x_vals]

• The function f_same = 1/x has an issue at $x = 0$ (division by zero), so we replace that with NaN (Not a Number) to avoid errors;
• For f_special, we know that $\lim_{\raisebox{-1pt}{$x \to 0$}}\frac{\raisebox{1pt}{$sin(x)$}}{\raisebox{-1pt}{$x$}} = 1$, so we manually assign $1$ when $x = 0$.

Plotting Horizontal Asymptotes

axs[1].axhline(0, color='blue', linestyle='dashed', linewidth=2, label='y = 0 (horizontal asymptote)')
axs[2].axhline(1, color='red', linestyle='dashed', linewidth=2, label=r"$y = 1$ (horizontal asymptote)")

• The function 1/x has a horizontal asymptote at $y = 0$;
• The function sin(x)/x approaches $y = 1$, so we add a dashed red line for visual clarity.