What this chapter is about
This chapter introduces you to calculus, one of the most powerful tools in all of mathematics. Calculus studies how quantities change, and it begins with two fundamental concepts: limits and derivatives. A limit describes what value a function approaches as its input gets closer and closer to some number, even if the function is not defined exactly at that number. Understanding limits is essential because derivatives are built on them.
A derivative measures the instantaneous rate of change of a function. When you travel from Delhi to Agra, your average speed tells you total distance divided by total time, but your speedometer shows your speed at each instant — that instant-by-instant speed is what a derivative captures. In this chapter, you learn how to compute limits using algebraic techniques, and then use limits to define and calculate derivatives of polynomial and trigonometric functions.
After studying this chapter, you should be able to evaluate limits of various functions, understand what it means for a function to have a derivative, and compute derivatives using first principles and standard rules. These skills form the foundation for differential calculus, which you will use extensively in physics, economics, and higher mathematics.
Key ideas
- The limit of f(x) as x approaches a, written lim x→a f(x), is the value that f(x) gets arbitrarily close to as x gets arbitrarily close to a, regardless of whether f(a) is defined.
- If direct substitution gives a form like 0/0, you must simplify the expression by factoring, rationalising, or using standard limit results before evaluating.
- Two important standard limits are: lim x→0 (sin x)/x = 1 (where x is in radians) and lim x→0 (1 − cos x)/x = 0.
- The derivative of f(x) at x = a is defined as lim h→0 [f(a + h) − f(a)]/h, provided this limit exists. This is called differentiation from first principles.
- Geometrically, the derivative f'(a) gives the slope of the tangent line to the curve y = f(x) at the point where x = a.
- The derivative of a sum is the sum of derivatives, and a constant factor can be taken outside the derivative: d/dx [c·f(x)] = c·f'(x).
- Derivatives of basic functions follow patterns: d/dx (x^n) = n·x^(n−1) for any real n, d/dx (sin x) = cos x, and d/dx (cos x) = −sin x.
Formulas and facts to remember
Limit rules (assuming both limits exist):
- lim x→a [f(x) + g(x)] = lim x→a f(x) + lim x→a g(x)
- lim x→a [f(x) · g(x)] = lim x→a f(x) · lim x→a g(x)
- lim x→a [f(x)/g(x)] = [lim x→a f(x)] / [lim x→a g(x)], provided denominator is not zero
Standard limits:
- lim x→a (x^n − a^n)/(x − a) = n·a^(n−1)
- lim x→0 (sin x)/x = 1 (x in radians)
- lim x→0 (tan x)/x = 1
- lim x→0 (1 − cos x)/x² = 1/2
Derivative formulas:
- d/dx (c) = 0, where c is a constant
- d/dx (x^n) = n·x^(n−1)
- d/dx (sin x) = cos x
- d/dx (cos x) = −sin x
- d/dx (tan x) = sec²x
Derivative rules:
- Sum rule: d/dx [f(x) + g(x)] = f'(x) + g'(x)
- Constant multiple: d/dx [c·f(x)] = c·f'(x)
- Product rule: d/dx [f(x)·g(x)] = f(x)·g'(x) + g(x)·f'(x)
- Quotient rule: d/dx [f(x)/g(x)] = [g(x)·f'(x) − f(x)·g'(x)] / [g(x)]²
Worked examples
Example 1: Evaluating a limit by factoring
Find lim x→3 (x² − 9)/(x − 3).
Direct substitution gives (9 − 9)/(3 − 3) = 0/0, which is indeterminate.
Factor the numerator: x² − 9 = (x − 3)(x + 3).
So (x² − 9)/(x − 3) = (x − 3)(x + 3)/(x − 3) = x + 3, for x ≠ 3.
Now take the limit: lim x→3 (x + 3) = 3 + 3 = 6.
Example 2: Using a standard trigonometric limit
Find lim x→0 (sin 4x)/(3x).
Rewrite to match the standard form: (sin 4x)/(3x) = (4/3) · (sin 4x)/(4x).
Let u = 4x. As x → 0, we have u → 0, and (sin u)/u → 1.
Therefore, lim x→0 (sin 4x)/(3x) = (4/3) · 1 = 4/3.
Example 3: Finding a derivative from first principles
Find the derivative of f(x) = x² + 2x using first principles.
By definition, f'(x) = lim h→0 [f(x + h) − f(x)]/h.
Compute f(x + h) = (x + h)² + 2(x + h) = x² + 2xh + h² + 2x + 2h.
So f(x + h) − f(x) = x² + 2xh + h² + 2x + 2h − x² − 2x = 2xh + h² + 2h = h(2x + h + 2).
Thus [f(x + h) − f(x)]/h = 2x + h + 2.
Taking the limit as h → 0: f'(x) = 2x + 2.
Common mistakes
- Substituting directly into 0/0 forms without simplifying first → always check if the result is indeterminate, then factorise or rationalise before substituting.
- Forgetting that (sin x)/x → 1 only when x is in radians → always work in radians for calculus involving trigonometric limits.
- Writing d/dx (cos x) = sin x (missing the negative sign) → remember d/dx (cos x) = −sin x.
- Applying the product rule incorrectly as f'(x)·g'(x) → the correct form is f(x)·g'(x) + g(x)·f'(x).
- Confusing the derivative (instantaneous rate) with the average rate of change → the derivative uses a limit as h → 0, not a finite change.
Quick revision
- A limit describes the value a function approaches, not necessarily the value it reaches.
- lim x→0 (sin x)/x = 1 is fundamental; memorise it.
- Derivative = lim h→0 [f(x + h) − f(x)]/h = slope of tangent = instantaneous rate of change.
- Power rule: d/dx (x^n) = n·x^(n−1).
- Product and quotient rules let you differentiate combinations of functions.