Limits and Derivatives Class 11 Maths Notes

Key Terms: Limits and Derivatives

Limit

The value that a function 'approaches' as the input 'approaches' some value. Represented as lim(x→a) f(x) = L.

Left-Hand Limit (LHL)

The value a function approaches as x approaches 'a' from the left side (values less than 'a'). Represented as lim(x→a⁻) f(x).

Right-Hand Limit (RHL)

The value a function approaches as x approaches 'a' from the right side (values greater than 'a'). Represented as lim(x→a⁺) f(x).

Existence of a Limit

A limit for a function f(x) at x=a exists if and only if the Left-Hand Limit equals the Right-Hand Limit, i.e., lim(x→a⁻) f(x) = lim(x→a⁺) f(x).

Derivative

The instantaneous rate of change of a function with respect to one of its variables. Geometrically, it represents the slope of the tangent line to the function's graph at a specific point.

First Principle of Derivative

The formal definition of a derivative, calculated as the limit of the average rate of change. The formula is: f'(x) = lim(h→0) [f(x+h) - f(x)] / h.

Indeterminate Forms

Expressions in limits that are not well-defined, such as 0/0, ∞/∞, ∞ - ∞. These require further manipulation (e.g., factorization, rationalization) to evaluate the limit.

Intuitive Understanding of Limits

The concept of a limit is the cornerstone of calculus. It helps us understand the behavior of a function near a point, even if the function is not defined at that exact point. Imagine you are walking along the graph of a function, y = f(x). A limit asks: 'As your x-coordinate gets closer and closer to a certain value 'a', what y-coordinate are you approaching?' The crucial idea is approaching. The limit doesn't care about what happens at x = a; it only cares about the trend in the immediate vicinity of 'a'.

For example, consider the function f(x) = (x² - 1) / (x - 1). If you try to calculate f(1), you get 0/0, which is undefined. However, we can ask what value f(x) approaches as x gets very close to 1. If we try x = 0.9, f(x) = 1.9. If we try x = 0.99, f(x) = 1.99. From the other side, if we try x = 1.1, f(x) = 2.1, and if x = 1.01, f(x) = 2.01. We can see that as x gets infinitesimally close to 1 from either side, f(x) gets infinitesimally close to 2. Therefore, we say the limit of f(x) as x approaches 1 is 2, written as lim(x→1) f(x) = 2. This distinction between the function's value at a point and its limit at a point is fundamental.

Standard Limits Formulas

• Algebraic Limit: lim(x→a) (xⁿ - aⁿ) / (x - a) = naⁿ⁻¹
• Trigonometric Limit 1: lim(x→0) sin(x) / x = 1 (where x is in radians)
• Trigonometric Limit 2: lim(x→0) (1 - cos(x)) / x = 0
• Trigonometric Limit 3: lim(x→0) tan(x) / x = 1
• Exponential Limit 1: lim(x→0) (eˣ - 1) / x = 1
• Exponential Limit 2: lim(x→0) (aˣ - 1) / x = logₑ(a)
• Logarithmic Limit: lim(x→0) log(1 + x) / x = 1
• Polynomial Function: For any polynomial p(x), lim(x→a) p(x) = p(a).
• Rational Function: For a rational function f(x) = p(x)/q(x), lim(x→a) f(x) = p(a)/q(a), provided q(a) ≠ 0.

How to Find a Derivative using the First Principle

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Worked Examples

• {"title":"Evaluating a Limit using Factorization","bodyMarkdown":"Problem: Evaluate lim(x→2) (x² - 4) / (x - 2).\n\nSolution:\n1. Direct substitution gives (2² - 4) / (2 - 2) = 0/0 (Indeterminate form).\n2. Factorize the numerator: x² - 4 = (x - 2)(x + 2).\n3. Rewrite the limit: lim(x→2) [(x - 2)(x + 2)] / (x - 2).\n4. Cancel the (x - 2) term: lim(x→2) (x + 2).\n5. Substitute x = 2: 2 + 2 = 4.\nAnswer: 4"}
• {"title":"Finding a Derivative using the First Principle","bodyMarkdown":"Problem: Find the derivative of f(x) = x² using the first principle.\n\nSolution:\n1. f(x) = x²\n2. f(x+h) = (x+h)² = x² + 2xh + h²\n3. Formula: f'(x) = lim(h→0) [f(x+h) - f(x)] / h\n4. Substitute: lim(h→0) [ (x² + 2xh + h²) - x² ] / h\n5. Simplify: lim(h→0) [2xh + h²] / h\n6. Factor out h: lim(h→0) [h(2x + h)] / h\n7. Cancel h: lim(h→0) (2x + h)\n8. Evaluate limit: 2x + 0 = 2x\nAnswer: f'(x) = 2x"}

Standard Derivatives to Memorise

• Power Rule: d/dx (xⁿ) = nxⁿ⁻¹
• Constant Rule: d/dx (c) = 0, where c is a constant.
• Trigonometric Functions: - d/dx (sin x) = cos x - d/dx (cos x) = -sin x - d/dx (tan x) = sec² x - d/dx (cot x) = -csc² x - d/dx (sec x) = sec x tan x - d/dx (csc x) = -csc x cot x
• Product Rule: d/dx [u(x)v(x)] = u'(x)v(x) + u(x)v'(x)
• Quotient Rule: d/dx [u(x)/v(x)] = [u'(x)v(x) - u(x)v'(x)] / [v(x)]²
• Constant Multiple Rule: d/dx [c f(x)] = c f'(x)

Common Exam Traps

Be careful with the Quotient Rule. A common mistake is getting the order of subtraction wrong in the numerator. Remember the mnemonic 'Lo d(Hi) - Hi d(Lo), over Lo-Lo': [v(x)u'(x) - u(x)v'(x)] / [v(x)]². Another trap is forgetting the negative sign when differentiating cos(x) and cot(x). For limits, always check for the 0/0 indeterminate form before applying factorization or rationalization. If direct substitution gives a defined value, that's your answer! Don't overcomplicate it. Lastly, when using trigonometric limit formulas like sin(x)/x, ensure the argument of the function and the denominator are identical. For sin(2x)/x, you must rewrite it as 2 * sin(2x)/(2x).

Quick Revision Check

• What is the necessary and sufficient condition for a limit to exist at a point x = a? The Left-Hand Limit (LHL) must be equal to the Right-Hand Limit (RHL) at x = a.
• State the product rule for differentiation. If f(x) = u(x)v(x), then f'(x) = u'(x)v(x) + u(x)v'(x).
• What is the value of lim(x→0) sin(ax) / bx? a/b. You can rewrite it as (a/b) [lim(ax→0) sin(ax)/ax] = (a/b) 1 = a/b.
• What is the derivative of f(x) = 10? Why? The derivative is 0, because the rate of change of a constant is always zero.

Frequently Asked Questions

Frequently Asked Questions