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Limits and continuity — Formula Sheet

5 formulas from Calculus I, each with what it means, when to use it and the trap it sets.

Direct substitution

x \to a lim f (x) = f (a) when f is continuous at a

What it says: If nothing breaks at a, the limit is just the value there.
When to reach for it: Always try this first. It settles the majority of limit questions in one line.
Watch out: Substitution that produces 0/0 has told you nothing — it is an indeterminate form, not an answer. Factor or rationalise, then substitute again.

x \to 2 lim (x^{2} + 1) = 5

Nothing is undefined at x = 2, so the value is the limit.

Cancelling a 0/0 form

x \to a lim \frac{( x - a ) g ( x )}{( x - a ) h ( x )} = x \to a lim \frac{g ( x )}{h ( x )}

What it says: The factor causing 0/0 cancels, because x approaches a without ever equalling it.
When to reach for it: Substitution gave 0/0 and the top and bottom are polynomials or contain a surd.
Watch out: You may cancel only a common factor, never a common term. (x−a) must divide the whole numerator.

x \to 3 lim \frac{x ^{2} - 9}{x - 3} = x \to 3 lim (x + 3) = 6

Factoring first is what makes the substitution legal.

Two-sided limit exists

x \to a lim f (x) = L ⟺ x \to a^{-} lim f (x) = x \to a^{+} lim f (x) = L

What it says: The limit exists only if the approach from the left and from the right agree.
When to reach for it: Piecewise functions, absolute values, and any graph with a jump.
Watch out: Each side can exist perfectly well on its own while the two-sided limit does not exist at all.

x \to 0^{-} lim \frac{∣ x ∣}{x} = - 1, x \to 0^{+} lim \frac{∣ x ∣}{x} = 1

The sides disagree, so the limit at 0 does not exist.

Rational limits at infinity

x \to \infty lim \frac{a _{n} x ^{n} + \dots}{b _{m} x ^{m} + \dots} = ⎩ ⎨ ⎧ 0 a_{n} / b_{m} \pm \infty n < m n = m n > m

What it says: Only the highest power on each side matters once x is large enough.
When to reach for it: A ratio of polynomials as x runs off to ±∞, and for finding horizontal asymptotes.
Watch out: Compare the degrees, not the leading coefficients. A huge coefficient on a lower degree still loses.

x \to \infty lim \frac{3 x ^{2} + 9 x}{5 x ^{2} - 1} = \frac{3}{5}

Equal degrees, so the answer is the ratio of leading coefficients.

Continuity at a point

f continuous at a ⟺ x \to a lim f (x) = f (a)

What it says: The function is defined there, the limit exists there, and the two are the same number.
When to reach for it: Choosing a constant that makes a piecewise function join up, or justifying substitution.
Watch out: All three conditions are required. A hole where the limit exists but f(a) is undefined is still a discontinuity.

f (x) = {x + k 3 x x < 1 x \geq 1 \Rightarrow k = 2

Setting the two branch values equal at x = 1 is what pins k down.

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