Limits and Derivatives (Class 11): Formulas and Solved Examples

In the Class 11 Limits and Derivatives chapter, a limit ( lim_{xto a}f(x) ) is the value ( f(x) ) approaches as x approaches a. The key standard limits are ( lim_{xto a}frac{x^n – a^n}{x – a} = na^{n-1} ), ( lim_{xto0}frac{sin x}{x} = 1 ) and ( lim_{xto0}frac{1 – cos x}{x} = 0 ). The derivative is defined from first principles as ( f'(x) = lim_{hto0}frac{f(x + h) – f(x)}{h} ), which gives ( frac{d}{dx}x^n = nx^{n-1} ), ( frac{d}{dx}sin x = cos x ) and ( frac{d}{dx}cos x = -sin x ).

This guide follows the Limits and Derivatives chapter of the NCERT Class 11 Mathematics textbook (also used by most state boards and for JEE preparation). It explains every idea in the chapter, lists the formulas you must know, and works through the standard question types step by step. Every answer has been checked with the step-by-step solver on this page.

1. What is a limit?

We write ( lim_{xto a}f(x) = L ) if ( f(x) ) gets as close as we like to L when x is close enough to a (but not equal to a).

• The left-hand limit ( lim_{xto a^-}f(x) ) uses x slightly less than a.
• The right-hand limit ( lim_{xto a^+}f(x) ) uses x slightly more than a.
• However, the limit exists only if both one-sided limits exist and are equal.

Example 1. ( f(x) = frac{x^2 – 4}{x – 2} ) is undefined at ( x = 2 ), but for ( x neq 2 ) it equals ( x + 2 ). So ( lim_{xto2}f(x) = 4 ). A limit is about what happens near a point, not at it.

Example 2: a limit that doesn’t exist. For ( f(x) = frac{lvert xrvert}{x} ), the left-hand limit at 0 is ( -1 ) and the right-hand limit is ( 1 ). They differ, so ( lim_{xto0}f(x) ) does not exist. The one-sided limit calculator computes each side.

2. Algebra of limits

If ( lim_{xto a}f(x) ) and ( lim_{xto a}g(x) ) exist:

Rule	Statement
Sum/difference	( lim(f pm g) = lim f pm lim g )
Constant multiple	( lim(kf) = klim f )
Product	( lim(fg) = lim fcdotlim g )
Quotient	( limfrac{f}{g} = frac{lim f}{lim g} ), if ( lim g neq 0 )

For polynomials and rational functions defined at a, the limit is just the value: ( lim_{xto1}(x^3 – 2x + 5) = 4 ).

3. Methods for evaluating limits

Method	Use when	Example
Direct substitution	the function is defined at a	( lim_{xto3}(2x + 1) = 7 )
Factorisation	you get ( frac00 ) with polynomials	( lim_{xto2}frac{x^2 – 4}{x – 2} = 4 )
Rationalisation	( frac00 ) with square roots	( lim_{xto0}frac{sqrt{1 + x} – 1}{x} = frac12 )
Standard limits	expressions like ( frac{x^n – a^n}{x – a} ) or ( frac{sin x}{x} )	see below

Example 3: rationalisation. Multiply top and bottom by ( sqrt{1 + x} + 1 ):

$$lim_{xto0}frac{sqrt{1 + x} - 1}{x} = lim_{xto0}frac{(1 + x) - 1}{x(sqrt{1 + x} + 1)} = lim_{xto0}frac{1}{sqrt{1 + x} + 1} = frac12$$

4. Standard limits

The power limit

$$lim_{xto a}frac{x^n - a^n}{x - a} = na^{n-1}$$

Example 4. ( lim_{xto2}frac{x^3 – 8}{x – 2} = 3(2)^2 = 12 ).

Example 5. ( lim_{xto2}frac{x^5 – 32}{x – 2} = 5(2)^4 = 80 ).

Trigonometric limits

$$lim_{xto0}frac{sin x}{x} = 1, qquad lim_{xto0}frac{tan x}{x} = 1, qquad lim_{xto0}frac{1 - cos x}{x} = 0$$

These hold with x in radians. The first is proved with a geometric squeeze argument; see the limit of sin x / x and the squeeze theorem.

Example 6. ( lim_{xto0}frac{sin 3x}{x} = lim_{xto0}3cdotfrac{sin 3x}{3x} = 3 ).

Example 7. ( lim_{xto0}frac{1 – cos x}{x^2} ). Use ( 1 – cos x = 2sin^2frac{x}{2} ):

$$lim_{xto0}frac{2sin^2(x/2)}{x^2} = lim_{xto0}frac12left(frac{sin(x/2)}{x/2}right)^2 = frac12$$

Example 8. ( lim_{xto0}frac{sin ax}{sin bx} = frac{a}{b} ), for example ( lim_{xto0}frac{sin 4x}{sin 2x} = 2 ).

Exponential and log limits (often added in the same unit)

$$lim_{xto0}frac{e^x - 1}{x} = 1, qquad lim_{xto0}frac{ln(1 + x)}{x} = 1$$

Check a limit step by step

Enter any limit; the solver shows the method. For example, the example is Example 7.

The full limit calculator also does one-sided limits and limits at infinity (see limits at infinity). Note that the solver often uses L’Hôpital’s rule, which isn’t part of the Class 11 syllabus; in an exam, use the standard limits above.

5. Derivatives from first principles

The derivative of f at x is

$$f'(x) = lim_{hto0}frac{f(x + h) - f(x)}{h}$$

It’s the slope of the tangent to the curve, and the instantaneous rate of change. Finding it directly from this definition is called differentiation from first principles (or the ab initio method).

Example 9: ( f(x) = x^2 ).

$$f'(x) = lim_{hto0}frac{(x + h)^2 - x^2}{h} = lim_{hto0}frac{2xh + h^2}{h} = lim_{hto0}(2x + h) = 2x$$

Example 10: ( f(x) = frac1x ).

$$f'(x) = lim_{hto0}frac{frac{1}{x + h} - frac1x}{h} = lim_{hto0}frac{-h}{hx(x + h)} = -frac{1}{x^2}$$

Example 11: ( f(x) = sin x ). Using ( sin A – sin B = 2cosfrac{A + B}{2}sinfrac{A – B}{2} ):

$$f'(x) = lim_{hto0}frac{2cosleft(x + frac{h}{2}right)sinfrac{h}{2}}{h} = lim_{hto0}cosleft(x + frac{h}{2}right)cdotfrac{sin(h/2)}{h/2} = cos x$$

The full argument is in the derivative of sin x guide.

6. Rules of differentiation

Rule	Formula
Power	( frac{d}{dx}x^n = nx^{n-1} )
Sum	( (u pm v)’ = u’ pm v’ )
Product (Leibniz)	( (uv)’ = u’v + uv’ )
Quotient	( left(frac{u}{v}right)’ = frac{u’v – uv’}{v^2} )
( sin x ), ( cos x ), ( tan x )	( cos x ), ( -sin x ), ( sec^2 x )

Example 12: a polynomial. ( frac{d}{dx}(x^3 – 4x^2 + 7) = 3x^2 – 8x ).

Example 13: product. ( frac{d}{dx}[(x + 1)(x – 2)] ). Either expand to ( x^2 – x – 2 ) and differentiate, or use the product rule: ( 1cdot(x – 2) + (x + 1)cdot1 = 2x – 1 ).

Example 14: quotient. ( frac{d}{dx}frac{x}{1 + x} = frac{(1 + x) – x}{(1 + x)^2} = frac{1}{(1 + x)^2} ).

Example 15: trig. ( frac{d}{dx}(xsin x) = sin x + xcos x ).

All the rules, with proofs and more examples, are in derivative rules, the product rule and the quotient rule guides.

7. Question types in exams

1. Evaluate a limit by substitution or factorisation. Always try substituting first; only factorise if you get ( frac00 ).
2. Use a standard limit. Rewrite the expression to match ( frac{x^n – a^n}{x – a} ) or ( frac{sin x}{x} ).
3. Left and right limits of piecewise functions. Find each side separately; the limit exists only if they match.
4. Find a constant so a limit exists. Set the left- and right-hand limits equal and solve.
5. Derivative from first principles. Write the definition, expand, cancel h, then let ( h to 0 ).
6. Derivative using rules. Simplify first, then apply product or quotient rules.

Example 16 (type 4). Find k if ( lim_{xto1}f(x) ) exists, where ( f(x) = kx + 3 ) for ( x < 1 ) and ( f(x) = 2x^2 ) for ( x ge 1 ). Left limit ( k + 3 ), right limit ( 2 ), so ( k = -1 ).

Common mistakes

• Writing ( frac00 = 0 ) or 1. ( frac00 ) means “not yet determined”: simplify further.
• Degrees in trig limits. ( frac{sin x}{x} to 1 ) only in radians.
• Cancelling h too early in first-principle questions, before expanding ( f(x + h) ).
• Using ( (uv)’ = u’v’ ). Use the product rule.
• Forgetting that a limit can exist where the function doesn’t. For example, Example 1 has a limit at a point where f is undefined.

Practice questions

1. ( lim_{xto3}frac{x^2 – 9}{x – 3} )
2. ( lim_{xto1}frac{x^{10} – 1}{x – 1} )
3. ( lim_{xto0}frac{sin 5x}{3x} )
4. ( lim_{xto0}frac{tan 2x}{x} )
5. Find the derivative of ( 3x^2 + 2x ) from first principles.
6. Differentiate ( frac{x^2 + 1}{x} ).

Answers: (1) 6; (2) 10; (3) ( frac53 ); (4) 2; (5) ( 6x + 2 ); (6) ( 1 – frac{1}{x^2} ).

FAQ

What is the formula for a derivative from first principles?

( f'(x) = lim_{hto0}frac{f(x + h) – f(x)}{h} ).

What are the standard limits in Class 11?

( lim_{xto a}frac{x^n – a^n}{x – a} = na^{n-1} ), ( lim_{xto0}frac{sin x}{x} = 1 ), ( lim_{xto0}frac{tan x}{x} = 1 ) and ( lim_{xto0}frac{1 – cos x}{x} = 0 ).

When does a limit not exist?

When the left-hand and right-hand limits differ, or when the function grows without bound near the point.

Is L’Hôpital’s rule allowed in Class 11?

It’s not part of the Class 11 syllabus, so exam answers should use factorisation, rationalisation and standard limits. It’s still a useful way to check your answer.

What comes after Limits and Derivatives?

In Class 12: continuity and differentiability, applications of derivatives, and integrals.

Further reading

• Limit of a function (Wikipedia) — formal and intuitive definitions with examples.
• Calculus I (Paul’s Online Math Notes) — the Limits and Derivatives chapters, covering the same material in more depth.