Limits and Derivatives

In this chapter: the idea of a derivative from speed, limits with left-hand and right-hand limits, algebra of limits, limits of polynomial and rational functions, the standard limit (xⁿ − aⁿ)/(x − a), trigonometric limits, the derivative from first principles, algebra of derivatives, and derivatives of polynomial and trigonometric functions.

Where the idea comes from

A stone dropped from a height falls s = 4.9t² metres in t seconds. Its average velocity between t = 2 and t = 2 + h is

As h shrinks towards 0, this gets as close as we like to 19.6 m/s. We cannot put h = 0 directly (that gives 0/0), but we can say the average velocity tends to 19.6 as h tends to 0. That number is the instantaneous velocity at t = 2. Making this precise is what limits are for.

Limits

We write lim_x→a f(x) = l if f(x) gets arbitrarily close to l as x gets close to a (from either side), whether or not f is defined at a.

• Left-hand limit (LHL): the value f(x) approaches as x → a through values less than a, written lim_x→a⁻ f(x).
• Right-hand limit (RHL): the value approached as x → a through values greater than a, written lim_x→a⁺ f(x).
• The limit exists exactly when the LHL and RHL both exist and are equal. That common value is the limit.

A classic example: for f(x) = |x|/x, the LHL at 0 is −1 and the RHL is 1, so lim_x→0 f(x) does not exist. For the greatest integer function, at any integer n the LHL is n − 1 and the RHL is n.

Algebra of limits

If lim f(x) and lim g(x) both exist as x → a, then the limit of f + g, f − g, fg and cf is the sum, difference, product and constant multiple of the limits. The limit of f/g is the quotient of the limits provided lim g(x) ≠ 0.

• For a polynomial, lim_x→a f(x) = f(a): just substitute.
• For a rational function f(x)/g(x): if g(a) ≠ 0, substitute. If substitution gives 0/0, factor out (x − a) from numerator and denominator, cancel, then substitute.

Worked example: Evaluate (a) lim_x→2 (x² − 4)/(x − 2), (b) lim_x→1 (x¹⁵ − 1)/(x¹⁰ − 1).
Solution: (a) Substitution gives 0/0. Factor: (x − 2)(x + 2)/(x − 2) = x + 2 for x ≠ 2, so the limit is 4.
(b) Divide numerator and denominator by (x − 1): the limit is [lim (x¹⁵ − 1)/(x − 1)] / [lim (x¹⁰ − 1)/(x − 1)] = (15 · 1¹⁴)/(10 · 1⁹) = 3/2.

Trigonometric limits

Using the sandwich theorem (if f(x) ≤ g(x) ≤ h(x) near a and f and h have the same limit l at a, then g also tends to l), NCERT proves two basic results. The angle x must be in radians.

From these: lim (tan x)/x = 1, lim (sin ax)/(bx) = a/b, and lim (1 − cos x)/x² = 1/2 (write 1 − cos x = 2 sin²(x/2)). For example, lim_x→0 (sin 4x)/(sin 2x) = lim [(sin 4x)/(4x)] × [(2x)/(sin 2x)] × 2 = 2.

JEE also uses three more standard limits that are not proved in this chapter: lim_x→0 (e^x − 1)/x = 1, lim_x→0 log_e(1 + x)/x = 1, and lim_x→0 (1 + x)^1/x = e.

Derivatives

The derivative of f at a is

Geometrically, [f(a + h) − f(a)]/h is the slope of the secant through P(a, f(a)) and Q(a + h, f(a + h)). As h → 0, Q slides along the curve to P and the secant becomes the tangent, so f′(a) is the slope of the tangent at P. Physically, it is the instantaneous rate of change of f.

Working out f′(x) straight from this definition is called differentiating from first principles. The derivative is also written dy/dx or d/dx f(x).

Worked example: From first principles, find the derivatives of (a) f(x) = x², (b) f(x) = 1/x, (c) f(x) = sin x.
Solution: (a) [(x + h)² − x²]/h = (2xh + h²)/h = 2x + h → 2x.
(b) [1/(x + h) − 1/x]/h = −h/[x(x + h)h] = −1/[x(x + h)] → −1/x² (x ≠ 0).
(c) [sin(x + h) − sin x]/h = 2 cos(x + h/2) sin(h/2)/h = cos(x + h/2) × [sin(h/2)/(h/2)] → cos x × 1 = cos x.

Algebra of derivatives

Standard derivatives

Worked example: Differentiate (a) x² sin x, (b) (x + 1)/(x − 1).
Solution: (a) Product rule: 2x sin x + x² cos x.
(b) Quotient rule: [(1)(x − 1) − (x + 1)(1)]/(x − 1)² = −2/(x − 1)², for x ≠ 1.

Common mistakes: (1) Treating 0/0 as 0 or 1; it is an indeterminate form, a signal to simplify first. (2) Using sin x/x → 1 with x in degrees. (3) Writing the quotient rule numerator as uv′ − u′v; the order matters. (4) Assuming the limit equals f(a); f(a) may not exist or may differ from the limit. (5) Declaring a limit to exist after checking only one side, which fails for |x|/x and [x]. (6) Writing (uv)′ = u′v′.

Exam focus

• 0/0 limits of rational functions by factorising, and the standard result for (xⁿ − aⁿ)/(x − a).
• Trigonometric limits using sin x/x → 1 and 1 − cos x = 2 sin²(x/2).
• One-sided limits of |x|, [x] and piecewise functions.
• Derivatives from first principles of simple functions.
• Product and quotient rules with polynomial and trigonometric functions.

Practice questions