Limits and continuity for NDA

Limits are the foundation of calculus, but in the NDA paper they are mostly quick evaluation questions. Almost every one falls to a small toolkit: substitute, factorise, rationalise, use a standard limit, or divide by the highest power. Continuity questions ask for the constant that makes a function continuous, or whether a function like |x| or [x] behaves well at a point. Learn the standard limits by heart and the chapter becomes one of the fastest in the paper.

Typical question types are:

• 0/0 limits that factorise or rationalise;
• trigonometric limits built on sin x/x;
• exponential and logarithmic limits;
• limits at infinity of rational functions;
• limits of the form 1 raised to ∞, which lead to e;
• whether a limit exists, by comparing left- and right-hand limits;
• finding k so that a function is continuous, and testing differentiability.

What a limit means

lim f(x) as x → a is the value f(x) approaches as x gets close to a, from both sides, whether or not f(a) itself is defined. The left-hand limit (x → a⁻) approaches from below, and the right-hand limit (x → a⁺) from above. The limit exists only when both are finite and equal.

For example, |x|/x equals −1 for x < 0 and 1 for x > 0. The two one-sided limits at 0 differ, so the limit does not exist.

Indeterminate forms are the ones where substitution tells you nothing: 0/0, ∞/∞, ∞ − ∞, 0 × ∞, 1 raised to ∞, 0⁰ and ∞⁰. A form like 5/0 is not indeterminate; it means the function grows without bound.

Standard limits

As x → 0 unless stated:

The angle in trigonometric limits is in radians. Two scaled versions come up constantly: sin kx/x → k and (1 − cos kx)/x² → k²/2.

The toolkit

1. Substitute. If you get a finite number, that is the limit.
2. Factorise a 0/0 form and cancel the common factor.
3. Rationalise when there is a square root: multiply top and bottom by the conjugate.
4. Reshape into a standard limit. For sin 3x/x, write 3 × sin 3x/(3x).
5. At infinity, divide by the highest power of x.
6. For 1 raised to ∞: if f → 1 and g → ∞, then lim f raised to g = e raised to lim g(f − 1).

L'Hôpital's rule (differentiate the top and bottom separately for 0/0 or ∞/∞) is a useful check if you know it, but every limit on this page can be done without it.

Rational functions at infinity. For a polynomial of degree p over one of degree q:

Continuity

f is continuous at x = a when three things hold: f(a) is defined, the limit as x → a exists, and the two are equal. In one line: lim f(x) as x → a = f(a).

• Polynomials, sin x, cos x, eˣ and |x| are continuous everywhere. ln x is continuous for x > 0, and a rational function wherever its denominator is not zero.
• Sums, products and compositions of continuous functions are continuous.
• The greatest integer function [x] jumps at every integer, so it is discontinuous there.
• A removable discontinuity is one where the limit exists but f(a) is missing or wrong. Redefining f(a) fixes it.

Differentiability at a is a stronger condition. The left-hand derivative lim (f(a + h) − f(a))/h as h → 0⁻ must equal the right-hand one as h → 0⁺. Differentiable implies continuous, but not the other way round. The standard example is |x| at 0: it is continuous, but the left derivative is −1 and the right derivative is +1, a sharp corner.

Worked NDA-style MCQs

Q1. lim (√(1 + x) − 1)/x as x → 0 is:
(a) 0 (b) 1 (c) 1/2 (d) 2

Multiply by √(1 + x) + 1 above and below: the top becomes (1 + x) − 1 = x, which cancels, leaving 1/(√(1 + x) + 1) → 1/2. Answer: (c).

Q2. lim (1 − cos 4x)/x² as x → 0 is:
(a) 4 (b) 8 (c) 16 (d) 2

1 − cos 4x = 2 sin²2x, so the expression is 2 × (sin 2x/x)² → 2 × 2² = 8. The scaled rule gives k²/2 = 16/2 directly. Answer: (b).

Q3. lim (1 + 3/x)²ˣ as x → ∞ is:
(a) e³ (b) e⁶ (c) √(e³) (d) 1

This is the 1 raised to ∞ form. e raised to lim 2x × (3/x) = e⁶. Answer: (b).

Q4. lim (e³ˣ − 1)/sin 2x as x → 0 is:
(a) 3/2 (b) 2/3 (c) 6 (d) 1

Write it as [(e³ˣ − 1)/(3x)] × [2x/sin 2x] × (3x/2x). The first two factors tend to 1, leaving 3/2. Answer: (a).

Q5. f(x) = sin 5x/(3x) for x ≠ 0 and f(0) = k is continuous at 0 when k is:
(a) 5 (b) 3/5 (c) 5/3 (d) 0

k must equal the limit, which is 5/3. Answer: (c).

Q6. f(x) = kx + 1 for x ≤ 2 and f(x) = 3x − 1 for x > 2. f is continuous at x = 2 when k is:
(a) 1 (b) 2 (c) 3 (d) 5/2

Left value 2k + 1; right limit 3(2) − 1 = 5. So 2k + 1 = 5 and k = 2. With k = 2 the left piece has slope 2 and the right piece slope 3, so f is continuous but not differentiable at 2. Answer: (b).

Common mistakes

• Using degrees in trigonometric limits. sin x/x → 1 only when x is in radians.
• Cancelling a factor that is not common to the whole numerator and denominator.
• Treating 1 raised to ∞ as 1. It is an indeterminate form, usually a power of e.
• Declaring a limit from one side. Check both sides at a jump or a modulus.
• Assuming continuous means differentiable. |x| at 0 is the standard counter-example.

Practice set

1. lim (x² − 1)/(x − 1) as x → 1 is: (a) 0 (b) 1 (c) 2 (d) does not exist
2. lim tan 2x/x as x → 0 is: (a) 1 (b) 2 (c) 1/2 (d) 0
3. lim (2x³ + x)/(5x³ − 1) as x → ∞ is: (a) 0 (b) 2/5 (c) 5/2 (d) ∞
4. lim (e²ˣ − 1)/x as x → 0 is: (a) 1 (b) 2 (c) e² (d) 1/2
5. lim (3ˣ − 1)/x as x → 0 is: (a) 3 (b) 1/3 (c) ln 3 (d) 0
6. lim (x² + 1)/(x³ + 2) as x → ∞ is: (a) 1 (b) 1/2 (c) ∞ (d) 0
7. lim x sin(1/x) as x → 0 is: (a) 1 (b) 0 (c) does not exist (d) ∞
8. At x = 2, the greatest integer function [x] is: (a) continuous and differentiable (b) continuous only (c) discontinuous (d) undefined

Answers:

1. (c). It reduces to x + 1.
2. (b). 2 × tan 2x/(2x) → 2.
3. (b). Equal degrees: the ratio of leading coefficients.
4. (b). 2 × (e²ˣ − 1)/(2x) → 2.
5. (c). The standard limit (aˣ − 1)/x → ln a.
6. (d). The denominator has the higher degree.
7. (b). |x sin(1/x)| ≤ |x|, which tends to 0.
8. (c). The left-hand limit is 1 and the right-hand limit is 2.

What to do next

• Write the standard-limits table from memory, including the scaled versions.
• Solve 25 old NDA limit and continuity questions at under a minute each, naming the tool before you start.
• Move straight on to differentiation, which is built on the limit definition, and then applications of derivatives.