2.4 Infinite limits and limits at infinity

Infinite limits and limits at infinity are crucial concepts in calculus. They help us understand how functions behave as they approach specific values or grow without bound. These ideas are key to grasping function behavior and asymptotes.

L'Hôpital's Rule is a powerful tool for evaluating tricky limits. It's especially useful for indeterminate forms, which often pop up when dealing with rational functions. Understanding these concepts helps us analyze function end behavior and identify asymptotes.

Infinite Limits and Limits at Infinity

Classification of infinite limits

• Occur when limit of function as x approaches specific value is infinity or negative infinity
  • Right-hand infinite limit $\lim_{x \to a^+} f(x) = \infty$ or $-\infty$ (function approaches infinity as x approaches a from the right)
  • Left-hand infinite limit $\lim_{x \to a^-} f(x) = \infty$ or $-\infty$ (function approaches infinity as x approaches a from the left)
• Vertical asymptotes exist at x-values where function approaches infinity or negative infinity
  • Identified by finding x-values that make denominator of rational function equal to zero (division by zero)

Function behavior at infinity

• Describe behavior of function as x approaches positive or negative infinity
  • $\lim_{x \to \infty} f(x) = L$ as x increases without bound, f(x) approaches value L
  • $\lim_{x \to -\infty} f(x) = L$ as x decreases without bound, f(x) approaches value L
• Horizontal asymptotes occur when function approaches specific y-value as x approaches positive or negative infinity
  • For rational functions $\frac{p(x)}{q(x)}$, where p(x) and q(x) are polynomials:
    1. If degree of p(x) < degree of q(x), horizontal asymptote is y = 0
    2. If degree of p(x) = degree of q(x), horizontal asymptote is y = $\frac{a_n}{b_m}$, where $a_n$ and $b_m$ are leading coefficients of p(x) and q(x)
    3. If degree of p(x) > degree of q(x), no horizontal asymptote exists

L'Hôpital's Rule for indeterminate forms

• Evaluate limits resulting in indeterminate forms: $\frac{0}{0}$, $\frac{\infty}{\infty}$, $0 \cdot \infty$, $\infty - \infty$, $0^0$, $1^\infty$, or $\infty^0$
• Apply L'Hôpital's Rule:
  1. Differentiate numerator and denominator separately
  2. Evaluate limit of new quotient
  3. Repeat process if new limit is still indeterminate form
• Applicable to one-sided limits and limits at infinity

End behavior of rational functions

• Refers to behavior of function as x approaches positive or negative infinity
• For rational functions:
  1. Determine degree of numerator and denominator polynomials
  2. Compare degrees to identify presence and value of horizontal asymptotes
  3. Evaluate limits at positive and negative infinity to confirm end behavior
• Oblique asymptotes occur when degree of numerator is exactly one more than degree of denominator
  • Slant line that function approaches as x approaches positive or negative infinity
  • Find equation of oblique asymptote by dividing numerator by denominator using long division and ignoring remainder