Study Notes: Asymptotic Behavior and Existence Theorems

Infinite Limits

Describe the behavior of functions as they grow without bound near a specific x-value.

Vertical Asymptote (VA)

A vertical line where the function approaches infinity as it approaches a specific x-value.

Limit Does Not Exist (DNE)

Occurs when a function grows without bound as x approaches a value.

One-Sided Limits

Limits that approach infinity from one side, either from the left (x→c-) or from the right (x→c+).

Non-Zero over Zero Rule

A vertical asymptote occurs at x=c if D(c)=0 and N(c)≠0.

Removable Discontinuity

Occurs when both the numerator and denominator of a function are zero at the same point.

Sign Analysis

A method to determine the direction of a limit by testing values near the asymptote.

Bottom Heavy Limit Case

When the degree of the denominator is greater than the numerator leading to a horizontal asymptote at y=0.

Balanced Limit Case

When the degrees of the numerator and denominator are equal, leading to a horizontal asymptote at y=an/bm.

Top Heavy Limit Case

When the degree of the numerator is greater than the denominator, resulting in no horizontal asymptote.

Dominance Hierarchy

Rank of functions by growth rate: exponentials, polynomials, logarithms, bounded functions.

Intermediate Value Theorem (IVT)

States that for any value between f(a) and f(b), there exists a c in (a, b) such that f(c)=k.

Continuity Condition for IVT

Necessary for using IVT; the function must be continuous on [a, b].

Different y-values Condition

For IVT, f(a) must not equal f(b) for k to exist between them.

Finding Roots with IVT

Apply IVT to show that a continuous function must cross the x-axis if it changes signs.

Common Mistake: VAs and Zeros

VA requires a non-zero numerator while D=0; 0/0 indicates a hole, not a VA.

Horizontal AS vs Vertical AS

Horizontal asymptotes are lines y=L. Vertical asymptotes are lines x=c.

Radical Sign Error Correction

For limits involving sqrt, always express as |x| to maintain sign integrity.

Conclusion from Sign Testing

sin(x)/x approaches 0 as x approaches infinity due to bounded function behavior.

EATS DC Memory Aid

Exponents Are The Same? Divide Coefficients for horizontal asymptotes.

Check for Continuity

Always verify continuity before applying IVT in proofs.

Behavior at Infinity

Limits at infinity describe end behavior as x approaches ±∞.

Function Growth Rates

Exponential functions grow faster than polynomial functions as x approaches infinity.

Potential for Slant Asymptotes

Occur when top degree is exactly one more than the bottom in rational functions.

Sign Behavior near Asymptotes

Testing values close to asymptotes determines if the graph approaches +∞ or -∞.

Oscillating Functions Limits

Oscillating functions generally approach zero when divided by growing functions.