Math · Calculus I · Concept
Limits at infinity
A limit at infinity asks what value f(x) approaches as x grows without bound, or falls without bound. If f(x) → L as x → ∞ or as x → −∞, the line y = L is a horizontal asymptote of the graph. For a rational function, dividing the numerator and denominator by the highest power of x in the denominator shows the answer: 0, the ratio of the leading coefficients, or unbounded growth, depending on the degrees.
What a limit at infinity means
lim f(x) = L as x → ∞ means that f(x) stays as close to L as we like once x is large enough. The graph levels off toward the line y = L, a horizontal asymptote, though it may still cross it. Limits as x → −∞ describe the left end in the same way.
The basic limit
As x → ∞, 1/xⁿ → 0 for every n > 0: a fixed numerator divided by an ever larger number shrinks toward 0. Every rational limit at infinity comes down to this fact.
Rational functions: divide by the highest power
Divide the numerator and the denominator by the highest power of x in the denominator. Every term left with x in its denominator tends to 0, and what remains is the limit.
| Degrees | Limit as x → ±∞ | Horizontal asymptote |
|---|---|---|
| Numerator lower | 0 | y = 0 |
| Equal | Ratio of the leading coefficients | y = that ratio |
| Numerator higher | Unbounded | None |
Square roots: watch the sign
√(x²) = |x|, not x. It equals x when x > 0 but −x when x < 0, so a function with a square root can approach different values at the two ends, as 2x/√(x² + 1) does.
Growth rates
Exponentials outgrow every power of x, and powers outgrow logarithms: xⁿ/eˣ → 0 and (ln x)/xᵖ → 0 as x → ∞, for any n and any p > 0. L’Hôpital’s rule proves these comparisons.
Differences of large terms
∞ − ∞ is not a number. When two large terms nearly cancel, as in √(x² + 4x) − x, multiply by the conjugate to turn the difference into a quotient, then divide by the highest power.
Common mistakes
- Substituting ∞ as if it were a number: ∞/∞ and ∞ − ∞ are indeterminate forms, not answers.
- Forgetting that √(x²) = |x|, which gives the wrong sign as x → −∞.
- Thinking a graph can never cross its horizontal asymptote: the asymptote describes only the ends.
- Dividing by the highest power in the numerator when the denominator’s is lower: use the denominator’s highest power.
Key terms
- Limit at infinity
- What f(x) approaches as x grows without bound (x → ∞) or falls without bound (x → −∞). It differs from an infinite limit, where the outputs grow without bound.
- Horizontal asymptote
- A horizontal line y = L that the graph approaches as x → ∞ or x → −∞. Unlike a vertical asymptote, the graph can cross it.
- End behavior
- How a function behaves as x tends toward positive or negative infinity. For a polynomial, the degree and leading coefficient determine the eventual directions of both ends.
- Indeterminate form
- A pattern such as 0/0 or ∞ − ∞ that substituting gives, which doesn’t decide the limit on its own. Different limits can give the same pattern, so more work is needed, such as factoring or L’Hôpital’s rule.
- Leading coefficient
- The coefficient of the highest-power term, such as 2 in 2x³ − x + 5. With the degree, it decides the graph’s end behavior.
Work through an example
Find the limit of (3x² − 5x + 1)/(2x² + 7) as x → ∞, and name the horizontal asymptote.
Find a limit at infinity of a rational function →Sources and scope
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Open the example inputs, change a value and keep a useful result on your board.
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