Evaluate limx→∞ (1 + 3/x)2x.
- Compare with the standard form lim (1 + a/x)bx = eab.
- Here a = 3 and b = 2, so ab = 3 × 2 = 6.
- Check by rewriting: [(1 + 3/x)x/3]6 → e6.
Answer: e6
Limits and continuity
NUST NET (Engineering) · Maths · Functions and limits. A short explanation of the idea, the rules to remember, the mistake to avoid, a worked example and practice questions with answers.
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
- limx→0 (sin x)/x = 1 and limx→0 (tan x)/x = 1, with x in radians; so limx→0 (sin ax)/(bx) = a/b
- limx→0 (1 − cos ax)/x2 = a2/2
- limn→∞ (1 + 1/n)n = e, and in general limn→∞ (1 + a/n)bn = eab
- limx→0 (ax − 1)/x = ln a, so limx→0 (ex − 1)/x = 1
- limx→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
Practice questions
Try each one, then open the answer.
1. f(x) = kx + 1 for x ≤ 2, and f(x) = 3x − 1 for x > 2. f is continuous at x = 2 when k equals
- A3
- B5/2
- C2
- D1
Show answer
Answer: C. The two pieces must meet at x = 2: 2k + 1 = 3(2) − 1 = 5, so k = 2. Forgetting the +1 (2k = 5) gives 5/2.
2. f(x) = (sin 3x)/x for x ≠ 0, and f(0) = k. The value of k that makes f continuous at x = 0 is
- A1/3
- B3
- C1
- D0
Show answer
Answer: B. (sin 3x)/x = 3 × (sin 3x)/(3x) → 3 as x → 0, so k = 3 makes f(0) equal to the limit.
3. If x° means x degrees, then limx→0 (sin x°)/x equals
- A1
- B180/π
- Cπ/180
- D0
Show answer
Answer: C. x° = πx/180 radians, so (sin x°)/x = (π/180) × sin(πx/180)/(πx/180) → π/180. The result (sin x)/x → 1 holds only when x is in radians.
Keep going
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