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Boundary Behavior Analyzer

Analyze function behavior at infinity, endpoints, and asymptotes

Select Function Type

Rational Exponential a^x Polynomial ax^n Log ln(x)

f(x)=(x+)/(x-)

Boundary Behavior

End behavior: limits at +/- infinity

Vertical asymptotes: denominator=0

Horizontal asymptotes: end behavior limits

Domain boundaries: function edges

Boundary behavior analysis examines what happens to a function at the edges of its domain. This includes end behavior (x->inf), vertical asymptotes, and any domain restrictions. Understanding boundary behavior is essential for accurate graphing and limit analysis.

⚠ Boundary behavior must be checked at ALL edges of the domain, not just infinity. This includes points where the function is undefined.

What Is Boundary Behavior?

Boundary behavior describes how a function behaves as x approaches the limits of its domain. For polynomials, the highest-degree term determines end behavior. For rational functions, compare degrees. Exponential functions either explode or decay at infinity.

Rational Ends

deg num < deg den: y=0. deg num = deg den: y=ratio. deg num > deg den: no HA (oblique).

Polynomial Ends

Leading term a*x^n dominates. Even degree: both ends same sign. Odd degree: opposite signs.

Exponential

a>1: as x->inf, f->inf; as x->-inf, f->0. 0<a<1: opposite. Always positive.

Logarithm

ln(x): as x->0+, f->-inf (VA). as x->inf, f->inf (slow). Domain x>0.

Teaching Example: f(x)=(2x+3)/(x-2). VA at x=2 (den=0, num=7!=0). As x->inf, f->2/1=2 (HA y=2). As x->-inf, f->2. Left side of VA: -> -inf. Right side: -> +inf.

Applications

Graphing Limits Calculus Asymptotes Domain Analysis

Frequently Asked Questions

What is boundary behavior?▼

How f behaves at domain edges: infinity, asymptotes, endpoints. Check limits at all boundaries.

Rational end behavior?▼

Compare degrees. Lower num deg -> y=0. Equal -> ratio of coefficients. Higher num deg -> no HA.

Exponential end behavior?▼

a>1: inf as x->inf, 0 as x->-inf. 0<a<1: 0 as x->inf, inf as x->-inf. Always positive.

Domain boundary?▼

ln(x): boundary at x=0 (VA). sqrt(x): boundary at x=0 (f(0)=0). Always check domain edges.

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