Limits and continuity

NUST NET Maths · Maths · Functions and limits. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

A limit is the value a function gets close to as x approaches a number. It does not matter what the function does exactly at that number. Always try direct substitution first. If you get a real number, that is the answer.

If you get 0/0, the expression is hiding a common factor or a standard limit. Factorise and cancel, rationalise a square root, or rewrite the expression so that it matches one of the standard limits below.

A function is continuous at x = a when its graph has no break there: the left-hand limit, the right-hand limit and the value f(a) are all equal.

Rules to remember

lim_x→0 (sin x)/x = 1 and lim_x→0 (tan x)/x = 1, with x in radians; so lim_x→0 (sin ax)/(bx) = a/b
lim_x→0 (1 − cos ax)/x² = a²/2
lim_n→∞ (1 + 1/n)ⁿ = e, and in general lim_n→∞ (1 + a/n)^bn = e^ab
lim_x→0 (a^x − 1)/x = ln a, so lim_x→0 (e^x − 1)/x = 1
lim_x→a (xⁿ − aⁿ)/(x − a) = n aⁿ⁻¹
As x→∞ in a fraction of polynomials, divide by the highest power of x; equal degrees give the ratio of the leading coefficients
f is continuous at x = a when the left-hand limit = the right-hand limit = f(a)

Common mistake

Writing 0/0 = 0 or 'does not exist' and stopping. 0/0 only tells you to simplify first; also remember that (sin x)/x → 1 works only when x is in radians.

Worked example

Evaluate lim_x→∞ (1 + 3/x)^2x.

Compare with the standard form lim (1 + a/x)^bx = e^ab.
Here a = 3 and b = 2, so ab = 3 × 2 = 6.
Check by rewriting: [(1 + 3/x)^x/3]⁶ → e⁶.

Answer: e⁶

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