Class 11 — Mathematics (NCERT)

Limits and Derivatives

Class 11

✓By the end of this lesson students will be able to understand the intuitive concept of a limit of a function.
✓By the end of this lesson students will be able to evaluate limits of functions using algebraic methods and standard formulas.
✓By the end of this lesson students will be able to understand the concept of the derivative as the instantaneous rate of change.
✓By the end of this lesson students will be able to calculate the derivative of a function from first principles.

Key concepts

Introduction to Limits

The concept of a limit is fundamental to calculus. When we write lim (x→a) f(x) = L, it means that as x gets closer and closer to 'a' (but not equal to 'a'), the value of f(x) gets closer and closer to 'L'. We examine the behaviour of the function in the neighbourhood of 'a'.

Left-Hand Limit (L.H.L.) and Right-Hand Limit (R.H.L.)

Left-Hand Limit (L.H.L.): lim (x→a-) f(x) means x approaches 'a' from values smaller than 'a'. Right-Hand Limit (R.H.L.): lim (x→a+) f(x) means x approaches 'a' from values larger than 'a'. For a limit to exist at x = a, the L.H.L. must be equal to the R.H.L. That is, lim (x→a) f(x) = L if and only if lim (x→a-) f(x) = L and lim (x→a+) f(x) = L.

Algebra of Limits

If lim (x→a) f(x) = L and lim (x→a) g(x) = M, then:\n1. lim (x→a) [f(x) ± g(x)] = L ± M\n2. lim (x→a) [f(x) ⋅ g(x)] = L ⋅ M\n3. lim (x→a) [k ⋅ f(x)] = k ⋅ L (where k is a constant)\n4. lim (x→a) [f(x) / g(x)] = L / M, provided M ≠ 0.

Important Standard Limits

These formulas are crucial for evaluating limits quickly and efficiently.

1. lim (x→a) (x^n - a^n) / (x - a) = n ⋅ a^(n-1) (for any rational n)\n2. lim (x→0) sin(x) / x = 1\n3. lim (x→0) tan(x) / x = 1\n4. lim (x→0) (e^x - 1) / x = 1\n5. lim (x→0) (a^x - 1) / x = log_e(a) (for a > 0)\n6. lim (x→0) log_e(1+x) / x = 1

Derivative from First Principles (Definition)

The derivative of a function f(x) with respect to x, denoted by f'(x) or dy/dx, represents the instantaneous rate of change of f(x) with respect to x. Geometrically, it gives the slope of the tangent to the curve y = f(x) at any point (x, f(x)). It is defined using limits.

f'(x) = lim (h→0) [f(x+h) - f(x)] / h

Key facts to remember

1A limit lim (x→a) f(x) exists if and only if lim (x→a-) f(x) = lim (x→a+) f(x).
2If direct substitution results in an indeterminate form (0/0, ∞/∞), algebraic manipulation (factorisation, rationalisation, using identities) is required.
3The derivative f'(x) represents the instantaneous rate of change of f(x) and the slope of the tangent to the curve y = f(x) at (x, f(x)).
4The definition of the derivative from first principles is f'(x) = lim (h→0) [f(x+h) - f(x)] / h.
5Memorise the standard limit formulas, especially lim (x→a) (x^n - a^n) / (x - a) = n ⋅ a^(n-1) and lim (x→0) sin(x) / x = 1.

Worked examples

Example 1

Evaluate lim (x→2) (x^2 - 4) / (x - 2).

IStep 1: Direct substitution of x = 2 gives (2^2 - 4) / (2 - 2) = 0 / 0, which is an indeterminate form. Hence, we need to simplify the expression.

IIStep 2: Factorise the numerator: x^2 - 4 = (x - 2)(x + 2).

IIIStep 3: Substitute this back into the limit expression: lim (x→2) [(x - 2)(x + 2)] / (x - 2).

IVStep 4: Since x → 2, x ≠ 2, so (x - 2) ≠ 0. We can cancel (x - 2) from the numerator and denominator.

VStep 5: The expression simplifies to lim (x→2) (x + 2).

VIStep 6: Now, substitute x = 2: 2 + 2 = 4.

Answer

Always check for indeterminate forms like 0/0 or ∞/∞ before applying algebraic simplification methods like factorisation or rationalisation.

Example 2

Evaluate lim (x→0) sin(3x) / x.

IStep 1: We know the standard limit formula: lim (θ→0) sin(θ) / θ = 1.

IIStep 2: To use this formula, the argument of the sine function in the numerator must match the denominator. Here, we have sin(3x) in the numerator and x in the denominator.

IIIStep 3: Multiply and divide the expression by 3 to match the form: lim (x→0) [sin(3x) / (3x)] ⋅ 3.

IVStep 4: Let θ = 3x. As x → 0, θ also approaches 0.

VStep 5: So, the expression becomes lim (θ→0) [sin(θ) / θ] ⋅ 3.

VIStep 6: Using the standard limit, lim (θ→0) sin(θ) / θ = 1. Therefore, the limit is 1 ⋅ 3 = 3.

Answer

Ensure the argument of the trigonometric function in the numerator exactly matches the denominator when applying standard limit formulas.

Example 3

Find the derivative of f(x) = x^2 from first principles.

IStep 1: Recall the definition of the derivative from first principles: f'(x) = lim (h→0) [f(x+h) - f(x)] / h.

IIStep 2: Substitute f(x) = x^2 into the formula. First, find f(x+h).

IIIStep 3: f(x+h) = (x+h)^2 = x^2 + 2xh + h^2.

IVStep 4: Substitute f(x+h) and f(x) into the formula: f'(x) = lim (h→0) [(x^2 + 2xh + h^2) - x^2] / h.

VStep 5: Simplify the numerator: f'(x) = lim (h→0) [2xh + h^2] / h.

VIStep 6: Factor out h from the numerator: f'(x) = lim (h→0) [h(2x + h)] / h.

VIIStep 7: Since h → 0, h ≠ 0. We can cancel h from the numerator and denominator: f'(x) = lim (h→0) (2x + h).

VIIIStep 8: Now, substitute h = 0 into the expression: 2x + 0 = 2x.

Answer

f'(x) = 2x

The crucial step is to simplify the expression [f(x+h) - f(x)] / h such that 'h' can be cancelled from the denominator before substituting h = 0.

Common mistakes

✗Incorrectly assuming f(a) is the limit lim (x→a) f(x). The limit describes the behaviour *near* 'a', not necessarily *at* 'a'.
✗Failing to simplify indeterminate forms before substituting the limit value, leading to incorrect answers.
✗Not ensuring the argument of the trigonometric function matches the denominator when using standard limits (e.g., treating lim (x→0) sin(2x)/x as 1 instead of 2).
✗Algebraic errors when expanding f(x+h) or simplifying [f(x+h) - f(x)] / h in first principles problems.
✗Forgetting to write 'lim (h→0)' at each step until 'h' is actually substituted, which is a common notation error.

Exam tips

★For limit problems, always try direct substitution first. If it gives a definite value, that's the limit. If it's an indeterminate form, proceed with algebraic simplification.
★When using standard limit formulas, ensure the expression perfectly matches the required form. Make necessary adjustments (e.g., multiplying and dividing by a constant).
★In derivative from first principles problems, be meticulous with algebraic expansion and simplification. The ultimate goal is to cancel 'h' from the denominator.
★Clearly show all steps, including the 'lim' notation, until the limit is finally evaluated to avoid losing marks.

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