As x grows without bound, only the biggest powers matter. Compare the degrees, and the end behaviour reads off in one glance.
The problem
limx→∞ 3x² + 5xx² + 2
Top and bottom have the same degree, so the limit is the ratio of the leading coefficients: 31 = 3. Divide everything by x² and watch the small terms die: (3 + 5x) ÷ (1 + 2x²) → 3 ÷ 1. The 5x looks big but is nothing beside 3x² once x is large.
Every term with an x underneath it shrinks to zero as x grows.
3x² + 5xx² + 2 = 3 + 5x1 + 2x²
divided through by x²
5x → 0, 2x² → 0
anything over a giant is nothing
limit = 31 = 3
only the leading terms survive
| Degrees | Limit as x → ∞ | Example |
|---|---|---|
| top < bottom | 0 | 2x + 1x² + 3 → 0 |
| top = bottom | ratio of leading coefficients | 3x² + 5xx² + 2 → 3 |
| top > bottom | ∞ or −∞ | x³ + 1x + 2 → ∞ |
Heavier bottom drags it to zero; heavier top blows it up; a tie splits the difference at the coefficients.
Backwards: if ax² − 15x² + x has the limit 2 as x → ∞, the degrees tie, so a5 = 2 and a = 10.
In 3x² + 5xx² + 2, the 5x has the biggest coefficient — and no vote at all. Rank terms by power, never by coefficient: at x = 1 000, the 3x² term is worth three million while 5x is worth five thousand. Coefficients only break the tie between equal powers.
A product on top: multiply only the leading terms. In (4x + 1)x − 22x² + 7 the top leads with 4x · x = 4x², degree 2 like the bottom, so the limit is 42 = 2.
Find the limit of √9x² + 52x − 1 as x → ∞, and then as x → −∞.
√9x² + 5 behaves like √9x² = 3|x|
the + 5 has no vote far out
x → ∞: 3x2x → 32
|x| = x for positive x
x → −∞: 3(−x)2x → −32
|x| = −x for negative x
2x³ + 1x² + 4 as x → −∞: like 2x → −∞
heavier top: follow the sign of x
The root applies to the coefficient too. And a square root is never negative, so for x → −∞ the root is −3x — the sign of the answer flips. With x → −∞, always ask which way the leading terms point before writing ∞.
Scrambled, with a minus. Find the limit of 6 − 3x³ + x2x³ + 5 as x → ∞.
top's highest power: −3x³
the 6 is written first but counts for nothing
degrees tie at 3
ratio of the x³ coefficients
limit = −32 = −1.5
the minus travels with the 3
What is different: reorder by power before reading anything. 65 (the constants) and 32 (sign lost) are the two wrong answers.
A huge coefficient on the lighter top. Find the limit of 1000x + 5x² − 3 as x → ∞.
degree 1 over degree 2
heavier bottom
÷ x²: 1000x + 5x²1 − 3x²
every top term has x below it
→ 01 = 0
the 1000 does not save it
What is different: at x = 1000 the value is still about 1, so the big-number test needs a bigger number here. At x = 1 000 000 it is about 0.001 ✓.
A big coefficient only delays the fall. Powers decide the limit; coefficients matter only when the powers tie.
A heavier top with a minus. Find the limit of 5 − x³x + 2 as x → ∞.
leading terms: −x³ over x
degree 3 beats degree 1
behaves like −x²
−x³ ÷ x
limit = −∞
x = 100: −999 995102 ≈ −9804 ✓
What is different: “heavier top” does not always mean +∞. The sign of the leading terms picks the direction.
Find, as x → ∞, the limits of (a) x² + 5xx + 2 − x and (b) √x² + 8x − x.
(a) [x² + 5x − x(x + 2)]/(x + 2)
one fraction, common denominator
= 3xx + 2 → 3
the x² terms cancel; equal degrees
(b) × [√x² + 8x + x] / [√x² + 8x + x]
conjugate, as in 00 limits
= x² + 8x − x²√x² + 8x + x = 8x√x² + 8x + x
the difference of squares kills the root
bottom ≈ x + x = 2x → 8x2x = 4
√x² + 8x behaves like x
Check (b) with x = 10 000: √100 080 000 ≈ 10 003.999, minus 10 000 gives about 4 ✓.
Most items are the degree rule in 5 seconds — rank by power, read the leading coefficients, and watch for a minus sign or an x → −∞. A difference of two growing terms is the UPCAT twist: combine or use the conjugate, never call it 0. A big-number test (x = 1000) confirms any answer.
The problem
limx→∞ 2x + 1x² + 3
The bottom's degree is higher — x² beats x — so the denominator grows far faster and the fraction is dragged to 0. A compares coefficients, but that only applies when the degrees TIE; D pairs the 2 with the wrong term entirely.
