Limits are crucial in modeling. Here you'll not only learn how to effectively use Python (along with helpful libraries sympy and matplotlib), but also how to evaluate, graph and analyze these graphs.

Defining the Functions

f_diff = (2 - x)  # Approaches 2 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 → 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 $\lim_{x \rarr 0}\frac{sin(x)}{x} = 1$, 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.