Limits, Continuity and Differentiation

Limits

lim_x→a f(x) = L means f(x) gets arbitrarily close to L as x approaches a (without needing f(a) to exist). The limit exists only if the left and right limits are equal.

Standard limits worth memorising:

L'Hôpital's rule: if f(x)/g(x) has the form 0/0 or ∞/∞ at a, then lim f/g = lim f′/g′ (if the latter exists). It can be applied repeatedly. Other indeterminate forms (0 · ∞, ∞ − ∞, 1^∞, 0⁰, ∞⁰) are first rewritten as a quotient, often by taking logarithms.

Example: lim_x→0 (tan x − sin x)/x³. Since tan x − sin x = tan x (1 − cos x), the limit is lim (tan x / x) · ((1 − cos x)/x²) = 1 · 1/2 = 1/2.

A quick numerical check (evaluating close to the limit point) is a useful sanity test:

import math

def near_zero(f, h=1e-4):
    return f(h)

print(f"sin x / x           -> {near_zero(lambda x: math.sin(x) / x):.6f}")
print(f"(1 - cos x) / x^2   -> {near_zero(lambda x: (1 - math.cos(x)) / x ** 2):.6f}")
print(f"(tan x - sin x)/x^3 -> {near_zero(lambda x: (math.tan(x) - math.sin(x)) / x ** 3, 1e-3):.6f}")
print(f"(1 + x)^(1/x)       -> {near_zero(lambda x: (1 + x) ** (1 / x), 1e-7):.6f}  (e = {math.e:.6f})")

sin x / x           -> 1.000000
(1 - cos x) / x^2   -> 0.500000
(tan x - sin x)/x^3 -> 0.500000
(1 + x)^(1/x)       -> 2.718282  (e = 2.718282)

Continuity and differentiability

• f is continuous at a if lim_x→a f(x) = f(a): the limit exists, f(a) exists, and they agree.
• f is differentiable at a if f′(a) = lim_h→0 (f(a + h) − f(a))/h exists (left and right derivatives agree).

Differentiable ⇒ continuous, but not the other way round: f(x) = |x| is continuous at 0, but its left derivative is −1 and its right derivative is +1, so it is not differentiable there (a corner).

Typical GATE question: for which constants is a piecewise function continuous or differentiable at the join? Equate the two pieces' values (continuity) and their derivatives (differentiability) at that point.

Mean value theorems

• Rolle's theorem: if f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then f′(c) = 0 for some c in (a, b).
• Lagrange's mean value theorem: under the same conditions without f(a) = f(b), there is a c in (a, b) with f′(c) = (f(b) − f(a)) / (b − a): somewhere the instantaneous slope equals the average slope.

Example: f(x) = x² on [1, 3]: the average slope is (9 − 1)/2 = 4 = 2c, so c = 2.

Maxima and minima

1. Find the critical points: where f′(x) = 0 or f′ does not exist.
2. Second derivative test: f″(c) < 0 ⇒ local maximum; f″(c) > 0 ⇒ local minimum; f″(c) = 0 ⇒ inconclusive (use the first-derivative sign change).
3. On a closed interval [a, b], the absolute maximum and minimum are among the critical points and the endpoints: evaluate f at all of them.

Example: f(x) = x³ − 6x² + 9x + 1. f′(x) = 3x² − 12x + 9 = 3(x − 1)(x − 3), and f″(x) = 6x − 12.

• At x = 1: f″ = −6 < 0, a local maximum, f(1) = 5.
• At x = 3: f″ = 6 > 0, a local minimum, f(3) = 1.
• On [0, 4]: f(0) = 1 and f(4) = 5, so the absolute maximum is 5 (at x = 1 and x = 4) and the absolute minimum is 1 (at x = 0 and x = 3).

For a function of two variables f(x, y): at a point where f_x = f_y = 0, let r = f_xx, s = f_xy, t = f_yy. If rt − s² > 0 it is a maximum (r < 0) or minimum (r > 0); if rt − s² < 0 it is a saddle point.

Practice questions