(373) Calculus 1 Lecture 3.5: Limits of Functions at Infinity

Limits at Infinity

• Discussing limits as X approaches positive or negative infinity.

• Recall: If limit as x approaches a number gives f(x) = ±∞ → it indicates an asymptote.

• Asymptote vs. removable discontinuity: if canceling out a limit leads to a point being defined, it’s a hole; otherwise, it’s an asymptote.

Types of Discontinuities

• Two primary cases:

  • Both sides approaching positive infinity: limit exists and equals positive infinity.

  • One side approaches positive, other side approaches negative infinity: limit does not exist.

• Discontinuities occur at points where the denominator is zero, which can lead to holes (removable) or asymptotes (non-removable).

Identifying Discontinuities in Rational Functions

• To find discontinuities, set the denominator equal to zero and solve:

  • Example with rational function; discontinuities at certain x-values depend on whether those values make the denominator zero.

• Holes: occur when both numerator and denominator are zero at the same point (can be canceled).

• Asymptotes: occur when the denominator is zero, but numerator is not, and cannot be simplified.

Case Study: Drawing Conclusions from Tests

• Performing sign analysis tests around discontinuities to determine limits approaching those points.

• Example provided illustrating reactions of a rational function at discontinuities (e.g. plugged-in values).

Horizontal Asymptotes

• Limits approaching infinity also predict horizontal behavior:

  • If a function approaches a constant as x approaches ±∞, a horizontal asymptote exists.

  • Horizontal asymptotes arise when examining limits at infinity.

Limits of Functions as x Approaches Infinity

• General behavior of polynomials:

  • Polynomials will either approach positive or negative infinity.

  • Leading term determines end behavior of polynomial.

Key Functions and Their Limits

• Functions like 1/x (as x approaches infinity):

  • Lessens toward zero, hence, horizontal asymtote at y=0.

• Any constant over a polynomial of higher degree goes to zero:

  • Example: Limit of 1/x² approaches 0 as x → ±∞.

Leading Terms in Polynomial Functions

• Calculation of limits motivated by dominating behaviors of the highest degree terms:

  • Example: Limit behavior of -3x³ indicates behavior of the polynomial as x approaches ± infinity.

Limit Computation Techniques

• When calculating limits,

  • Common practice: Divide numerator and denominator by the highest power in the denominator for clarity in asymptotic behavior.

Example Situations

• Conjugate multiplication for simplifying limits involving roots.

• Consider differences in behaviour when changing signs in polynomials vs rational expressions.

Special Cases and Rational Expressions

• Awareness of cases where direct plugging yields infinity: appropriate algebraic intervention is necessary to evaluate limits.

• Consistency in dividing terms by the dominant power in the expression helps elucidate final limit outputs.

Final Thoughts on Limit Behavior

• Always check for possible irrationalities or undefined expressions when evaluating limits approaching infinity.

• Understanding limits and asymptotic behaviors is crucial in calculus, with applications in determining function behaviors over unbounded domains.