1 · As x → a: factor and cancel
When direct substitution gives 0/0 in a rational function, the zero is a shared factor wearing a disguise. Factor numerator and denominator, cancel the common factor, then substitute.
Step 1: substitute: 0/0 — factor instead.
Step 2: numerator (x − 3)(x + 3); denominator (x − 3)(x + 2).
Step 3: cancel the (x − 3) — legal because the limit never touches x = 3 itself.
Step 4: lim x→3 (x + 3)/(x + 2) = 6/5.
If nothing cancels after full factoring, you are looking at a genuine infinite limit or a case where one side goes to ∞ and the other to −∞ (Section 5).
2 · Square roots: multiply by the conjugate
Radicals produce 0/0 through a difference of roots. The cure is the conjugate: multiply top and bottom by the mirror expression with the sign flipped, then use (A − B)(A + B) = A² − B².
Step 1: direct substitution: (2 − 2)/0 = 0/0.
Step 2: multiply by (√(x + 4) + 2)/(√(x + 4) + 2):
((x + 4) − 4) / (x·(√(x + 4) + 2)) = x / (x·(√(x + 4) + 2)).
Step 3: cancel x: 1/(√(x + 4) + 2).
Step 4: substitute x = 0: 1/(2 + 2) = 1/4.
3 · As x → ±∞: divide by the highest power
At infinity, no single term wins by magnitude except the highest-power one. Divide every term top and bottom by xⁿ (n = top degree, adjusting if needed) and watch the lower terms die as 1/x → 0.
Step 1: highest power is x² — divide: (2 + 3/x) / (5 − 7/x²).
Step 2: let x → ∞: 3/x → 0 and 7/x² → 0.
Step 3: lim = 2/5.
For radicals at infinity, pull the root out: √(9x² + 1) = |x|√(9 + 1/x²). Note the absolute value — for x → −∞, √(x²) = −x, not x.
4 · The degree comparison rule
For rational f(x) = p(x)/q(x) as x → ±∞, compare degrees:
- deg p < deg q → limit is 0 (horizontal asymptote y = 0).
- deg p = deg q → limit is the ratio of leading coefficients (horizontal asymptote y = a/b).
- deg p > deg q → the limit is ±∞; by exactly 1 degree the asymptote is a slant line, otherwise it is parabolic/cubic growth.
This rule is a shortcut, not a substitute for understanding — but at speed, it reads the answer off the top line of the problem.
5 · Absolute values and one-sided infinite limits
Absolute values split the domain at their zero. |x − a| equals (x − a) when x ≥ a and −(x − a) when x < a. So a limit involving |x − 1| near 1 must be done twice, once from each side.
Vertical asymptotes as one-sided limits: if lim x→a⁺ f(x) = ∞ and lim x→a⁻ f(x) = ∞, write lim x→a f(x) = ∞. If one side is ∞ and the other −∞, the two-sided limit does not exist (but the vertical asymptote x = a still stands). Denominator → 0 with numerator → nonzero is the signature of an infinite limit: check the sign on each side.
6 · Poly-type functions and the algebra behind them
"Poly-type" limits wrap polynomials in roots, absolute values or compositions. The meta-strategy never changes: (1) try substitution; (2) if 0/0, factor or conjugate; (3) if ∞/∞, divide by the highest power; (4) if |·| is present, split into cases; (5) if stuck, check whether both sides or both infinities agree.
Radical difference: multiply by conjugate A + B, use (A − B)(A + B) = A² − B²
∞/∞: divide every term by the highest power of x
deg p < deg q ⇒ 0 · deg p = deg q ⇒ a/b · deg p > deg q ⇒ ±∞
|x − a| = (x − a) if x ≥ a, and −(x − a) if x < a
√(x²) = |x| — minus sign applies when x → −∞
Common mistakes
- Canceling (x − 3) when x = 3 is exactly where the original is undefined without noting the hole — the shortcut is fine for limits, but say why it is legal.
- Multiplying by the conjugate of only the numerator and not the denominator (or vice versa).
- Writing √(x²) = x for negative x — it is |x|, and it flips sign at −∞.
- Declaring an infinite limit exists because the absolute value blows up, while the signed sides go to opposite infinities.
- Using the degree rule when the expression is a sum or difference, not a quotient — it applies to fractions only.
- Forgetting that |x − a| limits need a left case and a right case.