Infinite Limits and Asymptotes

For rational functions $\frac{p(x)}{q(x)}$:

• Degree(p) < Degree(q): $\lim = 0$. HA: $y = 0$.
• Degree(p) = Degree(q): $\lim = \frac{\text{leading coefficients}}{}$.
• Degree(p) > Degree(q): no HA (limit is $\pm\infty$).

When the denominator → 0 and numerator → nonzero: $\lim = \pm\infty$.

$\lim_{x \to 0^+} \frac{1}{x} = +\infty$. $\lim_{x \to 0^-} \frac{1}{x} = -\infty$.

$\lim_{x \to \infty} e^{-x} = 0$. $\lim_{x \to \infty} \ln x = \infty$ (slowly).