Limits at infinity and asymptotes help us understand how functions behave as x gets really big or small. We can figure out if a function levels off, grows forever, or approaches a slanted line. This knowledge is key for sketching graphs and predicting long-term trends.

By evaluating these limits, we can spot horizontal and oblique asymptotes. These invisible lines guide a function's shape as it stretches towards infinity. Understanding end behavior and asymptotes gives us a powerful toolkit for analyzing and visualizing complex functions.

Limits and Asymptotes

Limits at infinity

• Describe the behavior of a function as x becomes arbitrarily large or small
  • As x approaches positive infinity ($x \to \infty$), function values approach a specific value or infinity (exponential growth)
  • As x approaches negative infinity ($x \to -\infty$), function values approach a specific value or infinity (exponential decay)
• Evaluate limits at infinity for rational functions by dividing both numerator and denominator by the highest power of x in the denominator and simplifying the expression to determine the limit (polynomial long division)
• For exponential functions, as x approaches infinity:
  • If base is greater than 1, limit approaches infinity (rapid growth)
  • If base is between 0 and 1, limit approaches 0 (rapid decay)
• For logarithmic functions, as x approaches infinity, limit approaches infinity (slow growth)
• Use l'Hôpital's rule to evaluate limits of indeterminate forms when other methods fail

Horizontal and oblique asymptotes

• Horizontal asymptotes occur when the limit of a function as x approaches positive or negative infinity is a constant value
  • For rational functions, compare degrees of numerator and denominator polynomials:
    • If degree of numerator is less than degree of denominator, horizontal asymptote is y = 0
    • If degrees are equal, horizontal asymptote is y = ratio of leading coefficients
  • For exponential and logarithmic functions, no horizontal asymptotes exist
• Oblique (slant) asymptotes occur when the limit of a function as x approaches infinity is a linear function
  • For rational functions, if degree of numerator is one greater than degree of denominator, an oblique asymptote exists
  • Find equation of oblique asymptote by dividing numerator by denominator using long division and taking quotient without remainder ($y = mx + b$)
• Analyze continuity and differentiability at asymptotes to understand function behavior

End behavior of functions

• Describes how a function behaves as x approaches positive or negative infinity
• For polynomial functions:
  • If leading term has even degree and positive coefficient, function values approach positive infinity as x approaches both positive and negative infinity (U-shaped)
  • If leading term has even degree and negative coefficient, function values approach negative infinity as x approaches both positive and negative infinity (inverted U-shaped)
  • If leading term has odd degree and positive coefficient, function values approach positive infinity as x approaches positive infinity and negative infinity as x approaches negative infinity (increasing)
  • If leading term has odd degree and negative coefficient, function values approach negative infinity as x approaches positive infinity and positive infinity as x approaches negative infinity (decreasing)
• For rational functions, end behavior is determined by horizontal or oblique asymptotes
• For exponential functions, end behavior depends on base (growth or decay)
• For logarithmic functions, end behavior is function values approaching infinity as x approaches infinity (slow growth)

Sketching functions with limits

To sketch a function graph:

1. Determine domain of function
2. Find intercepts, if any (x and y intercepts)
3. Identify symmetry (even, odd, or periodic)
4. Calculate limits at infinity to determine horizontal or oblique asymptotes
5. Find first and second derivatives to determine intervals of increase/decrease and concavity (critical points and inflection points)
6. Identify local maxima, minima, or inflection points
7. Sketch graph using gathered information, paying attention to end behavior and asymptotes (connect key points and features)
8. Use graphical interpretation to visualize limits and asymptotes

Techniques for evaluating limits

• Direct substitution (when function is continuous at the point)
• Factoring and simplifying
• Rationalization (for limits involving radicals)
• Algebraic manipulation to rewrite the function in a more manageable form
• L'Hôpital's rule for indeterminate forms
• Graphical analysis to estimate or confirm limit values