Limits at Infinity and End Behavior

End Behavior and Limits at Infinity

Key idea: use end behavior (limits at infinity) to understand horizontal asymptotes and how functions behave as x grows without bound.
Horizontal asymptote: a horizontal line y = L that the graph approaches as x → ±∞, i.e., lim_{x→±∞} f(x) = L (when this limit exists).

Infinite limit at infinity: when a function grows without bound as x grows without bound.
- If lim{x→∞} f(x) = ∞ or lim{x→∞} f(x) = -∞, we say f has an infinite limit at ∞.
- If lim_{x→-∞} f(x) = ∞ or -∞, we say f has an infinite limit at -∞.
End behavior analogy: similar to what we know about polynomials, but considered for all kinds of functions.
Examples of infinite limits at infinity:
- lim_{x→∞} x = ∞
- lim_{x→∞} e^{x} = ∞
- lim_{x→-∞} x^{2} = ∞
- lim_{x→-∞} x^{3} = -∞
One over x^n tends to 0 as x → ±∞:
- lim_{x→±∞}
  \frac{1}{x^{n}} = 0 \, (n > 0)
Consequence: no horizontal asymptote exists for infinite limits in the direction where the function grows without bound, but a horizontal asymptote may exist in the opposite direction if that limit is finite.
End behavior intuition: the end behavior of polynomials is guided by the leading term a_n x^n.

General polynomial: P(x) = an x^{n} + a{n-1} x^{n-1} + fer … + a1 x + a0, where a_n 0 and n terminant degree.
Leading term dominates as |x| → ∞.
If n is even:
- If a_n > 0: as x → ±∞, P(x) → ∞.
- If a_n < 0: as x → ±∞, P(x) → -∞.
If n is odd:
- If an > 0: lim{x→∞} P(x) = ∞ and lim_{x→-∞} P(x) = -∞.
- If an < 0: lim{x→∞} P(x) = -∞ and lim_{x→-∞} P(x) = ∞.
Example interpretation (lead term dominates):
- For P(x) = an x^{n} + n x^{n-1} + …, the end behavior is determined by a_n and n.
Special case: 1/x^n -> 0 as x → ±∞, giving horizontal asymptote y = 0 for those reciprocal terms.
Important rule:
- lim{x→±∞} P(x) is governed by the leading term an x^{n} (the rest become negligible after factoring x^n).

General technique: evaluate limits at infinity by dividing every term by the highest power of x in the denominator.
Why divide by the highest power in the denominator:
- It eliminates the x-dependence in terms that would otherwise blow up or be undefined at infinity.
- It leaves a form where you can apply limit laws term-by-term.
Example 1: lim_{x→∞} \frac{3x+2}{x^{2}-1}
- Divide numerator and denominator by x^{2}:
  \lim{x\to\infty} \frac{\frac{3x}{x^{2}} + \frac{2}{x^{2}}}{1 - \frac{1}{x^{2}}} = \lim{x\to\infty} \frac{\frac{3}{x} + \frac{2}{x^{2}}}{1 - \frac{1}{x^{2}}}
- As x → ∞, 1/x → 0 and 1/x^2 → 0, so the limit becomes 0/1 = 0.
Example 2: lim{x→∞} \frac{5}{x^{2}} = 5 \cdot lim{x→∞} \frac{1}{x^{2}} = 0.
Example 3: lim{x→∞} \frac{-2}{x^{3}} = -2 \cdot lim{x→∞} \frac{1}{x^{3}} = 0.
Example 4 (constants): lim_{x→∞} 18 = 18.
Key takeaway for rationals:
- If deg(numerator) < deg(denominator): horizontal asymptote y = 0.
- If deg(numerator) = deg(denominator): horizontal asymptote y = (leading coefficient of numerator) / (leading coefficient of denominator).
- If deg(numerator) > deg(denominator): no horizontal asymptote; possible slant (oblique) asymptote if deg(numerator) = deg(denominator) + 1 (found via long division).

Consider a rational function where the numerator and denominator have the same degree, e.g.,
f(x) = \frac{8x^{2} + \dots}{2x^{2} + \dots}
Then the horizontal asymptote is y = \frac{8}{2} = 4.
A more explicit demonstration (divide by x^2):
$$\lim{x\to\infty} \frac{8x^{2} + \dots}{2x^{2} + \dots} = \lim{x\to\infty} \frac{8 + \dots/x^{2}}{2 + \dots/x^{2}} = \frac{8}{2} = 4.$n
Relating to a specific example from the transcript:
- If you have a leading ratio of -8 (numerator) to 2 (denominator), the horizontal asymptote is y = -8/2 = -4.
- To verify, divide by the highest power (x^4 in that example) and let x → ∞; all terms with 1/x^k vanish, leaving the constant ratio as the limit.

When testing end behavior, remember:
- For polynomials: leading term drives the end behavior; constants and lower-degree terms vanish in comparison as |x| → ∞.
- For rational functions: compare degrees to determine horizontal asymptotes; do the division by the highest power to compute the exact limit if degs are equal.
Direction matters for horizontal asymptotes in some non-polynomial cases; for simple polynomials, end behavior is symmetric when the degree is even and different on each end when the degree is odd.
Slant (oblique) asymptotes occur only when the degree of the numerator is exactly one more than the degree of the denominator; they can be found via long division giving a linear asymptote y = mx + b.
Connection to derivative motivation: understanding end behavior provides intuition for growth rates and can motivate derivative-based analysis in more advanced topics.

Polynomials f(x) = a_n x^{n} + …
- If n even:
- a_n > 0: f(x) → ∞ as x → ±∞; end behavior same on both ends.
- a_n < 0: f(x) → -∞ as x → ±∞; end behavior same on both ends.
- If n odd:
- a_n > 0: f(x) → ∞ as x → ∞; f(x) → -∞ as x → -∞.
- a_n < 0: f(x) → -∞ as x → ∞; f(x) → ∞ as x → -∞.
Rational functions N(x)/D(x):
- If deg(N) < deg(D): horizontal asymptote y = 0.
- If deg(N) = deg(D): horizontal asymptote y = (leading coeff of N) / (leading coeff of D).
- If deg(N) > deg(D): no horizontal asymptote; if deg(N) = deg(D) + 1, slant asymptote exists (found by long division).
Limits to compute by division:
- Example: lim_{x→∞} \frac{3x+2}{x^{2}-1} = 0, via dividing by x^2.
- Example: lim_{x→∞} \frac{5}{x^{2}} = 0.
Classic limits:
- lim_{x→∞} x = ∞
- lim_{x→∞} e^{x} = ∞
- lim_{x→-∞} x^{2} = ∞
- lim_{x→-∞} x^{3} = -∞
Simple rule: lim_{x→±∞} \frac{1}{x^{n}} = 0 for n > 0.

End behavior and horizontal asymptotes are determined by leading terms and degree comparisons, especially for polynomials and rational functions.
For rational functions, dividing by the highest power of x in the denominator is a standard, reliable method to evaluate limits as x → ±∞.
The ratio of leading coefficients determines horizontal asymptotes when degrees are equal; otherwise, the asymptote is y = 0 (or none) depending on degree comparison.
These tools align with the broader intuition about end behavior and set the stage for more advanced topics like slant asymptotes and derivative-based analysis.