Class 11 Maths · Original EZT practice
Limits and Derivatives (Class 11 Chapter 12): summary and practice
Limits and Derivatives starts calculus: what a limit means (approaching, not reaching), the algebra of limits, the two standard trigonometric limits, and the derivative defined as a limit through the first principle. All questions are original EZT tutor practice following the chapter's own section order.
6 original worked examples, 6 fresh self-check questions and 4 mixed questions, with explained answers. These are not copied textbook exercises, an official chapter index or a full answer key. No affiliation with NCERT is claimed.
Official sources for topic and edition context, checked 22 October 2026:
1. The idea of a limit
A limit is the value a function APPROACHES as x nears a point - not the value at the point. Left-hand and right-hand limits must agree for the limit to exist.
2. Algebra of limits
Limits split over sums, products and quotients (denominator's limit nonzero). A 0/0 form usually lifts after factoring and cancelling.
3. Trigonometric limits
Two standards as x tends to 0: sin x / x tends to 1, and (1 - cos x) / x tends to 0. Everything else in the section is built from these.
4. Derivatives from first principle
The derivative is the limit of [f(x + h) - f(x)] / h as h tends to 0. From it flow the rules: derivative of x^n is n x^(n-1), of sin x is cos x, of cos x is -sin x.
Choose, solve, check, then try a fresh question.
Before calculating, read the method prompt and name the condition that makes it useful. Write your own attempt. Open the steps only when you need them, and use the check to catch a wrong sign, unit or interpretation. Finish with the fresh question before reading its answer.
Lift a 0/0 form by factoring
Choose the method: Factor, cancel the zero factor, then substitute.
EZT practice question
Find the limit of (x^2 - 4)/(x - 2) as x tends to 2.
Show the worked solution
- Direct substitution gives 0/0 - no information yet.
- Factor: x^2 - 4 = (x - 2)(x + 2).
- Cancel (x - 2): the limit becomes the limit of (x + 2).
Answer: 2 + 2 = 4.
Check: Near x = 2 the function behaves exactly like x + 2 - at 1.9 it gives 3.9, at 2.1 it gives 4.1, closing on 4.
Watch for this: Do not stop at 0/0 and write 0 or undefined - factor first, the limit usually exists.
Now try it yourself
Limit of (x^2 - 9)/(x - 3) as x tends to 3?
Check your answer
6.
Cancels to x + 3.
Use the standard sin x / x limit
Choose the method: As x tends to 0, sin x / x tends to 1.
EZT practice question
Find the limit of sin 3x / x as x tends to 0.
Show the worked solution
- Write sin 3x / x = 3 x sin 3x / (3x).
- As x tends to 0, so does 3x.
- sin 3x / (3x) tends to 1.
Answer: 3 x 1 = 3.
Check: The coefficient always survives: sin kx / x tends to k - here k = 3.
Watch for this: sin 3x / x is not 1 - match the denominator to the angle first.
Now try it yourself
Limit of sin 5x / x as x tends to 0?
Check your answer
5.
Same rescaling.
Differentiate from first principle
Choose the method: Compute the limit of [f(x + h) - f(x)] / h as h tends to 0.
EZT practice question
From first principle, find the derivative of x^3.
Show the worked solution
- f(x + h) - f(x) = (x + h)^3 - x^3 = 3x^2 h + 3x h^2 + h^3.
- Divide by h: 3x^2 + 3x h + h^2.
- As h tends to 0, the h terms vanish.
Answer: 3x^2 - so at x = 2 the derivative is 12.
Check: Matches the power rule: derivative of x^n is n x^(n-1), here 3x^2.
Watch for this: Divide by h BEFORE taking the limit - skipping that leaves a 0/0 with no cancellation.
Now try it yourself
Derivative of x^4 from the power rule, at x = 1?
Check your answer
4.
4x^3 at x = 1.
Differentiate sums term by term
Choose the method: The derivative of a sum is the sum of the derivatives; constants ride along.
EZT practice question
Differentiate 5x^2 + sin x.
Show the worked solution
- Derivative of 5x^2 is 5 x 2x = 10x.
- Derivative of sin x is cos x.
- Add.
Answer: 10x + cos x.
Check: Each rule applied to its own term - the sum rule licenses exactly that.
Watch for this: The constant 5 multiplies the derivative; it does not become 5x^2's derivative 2x + 5.
Now try it yourself
Differentiate 3x^2 + cos x.
Check your answer
6x - sin x.
Derivative of cos x is -sin x.
Limit versus value
Choose the method: The limit describes approach; the function need not equal it, or even be defined, at the point.
EZT practice question
Is sin x / x equal to 1 at x = 0?
Show the worked solution
- At x = 0 the expression is 0/0 - undefined.
- Its LIMIT as x tends to 0 is 1.
- A limit is about the journey, not the destination.
Answer: No - it is undefined at 0, but its limit there is 1.
Check: This is exactly why the chapter defines derivatives through limits: the difference quotient is 0/0 at h = 0, yet its limit is a clean number.
Watch for this: Saying sin x / x equals 1 at 0 confuses the limit with the value.
Now try it yourself
If f(x) = x + 2 for all x except 2, and f(2) is undefined, what is the limit as x tends to 2?
Check your answer
4.
The limit ignores the point itself.
Use the (1 - cos x) / x limit
Choose the method: As x tends to 0, (1 - cos x) / x tends to 0.
EZT practice question
Find the limit of (1 - cos x) / x as x tends to 0.
Show the worked solution
- Multiply top and bottom by (1 + cos x): (1 - cos^2 x) = sin^2 x.
- Expression becomes (sin x / x) x (sin x / (1 + cos x)).
- First factor tends to 1, second to 0/2 = 0.
Answer: 1 x 0 = 0.
Check: The two standard limits multiply out exactly this way - the derivation uses only sin x / x.
Watch for this: Do not quote 1 for this one - the 1 belongs to sin x / x.
Now try it yourself
Limit of (1 - cos x) / x^2 as x tends to 0 (harder)?
Check your answer
1/2.
Same trick leaves sin^2 x / (x^2 (1 + cos x)) tending to 1/2.
Try a mixed set
Choose a method or explanation before opening each answer.
1. Limit of sin x / x as x tends to 0?
Show answer and reason
1.
The standard trigonometric limit.
2. Limit of (1 - cos x) / x?
Show answer and reason
0.
The second standard limit.
3. Derivative of x^n?
Show answer and reason
n x^(n-1).
From first principle.
4. Derivative of cos x?
Show answer and reason
-sin x.
Sign flips - the one that surprises.
Try it yourself
Work these on paper before checking any notes. Answers are deliberately not shown here.
- Limit of (x^2 - 9)/(x - 3) as x tends to 3.
- Derivative of x^4, evaluated at x = 1.
- Limit of sin 5x / x as x tends to 0.
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Questions about this practice page
- What is the difference between a limit and a value?
- The value is what the function equals AT the point; the limit is what it approaches nearby. sin x / x is undefined at 0 but its limit is 1 - the limit never asks what happens exactly at the point.
- Why is the derivative defined as a limit?
- The slope between two points is [f(x + h) - f(x)] / h, but at h = 0 that quotient is 0/0. The limit as h tends to 0 captures the slope the secants settle into - the tangent slope - without ever dividing by zero.
- How do I handle a 0/0 limit?
- Rewrite first: factor and cancel, or use a standard limit. Direct substitution only works when the function is defined and continuous at the point; 0/0 is a signal to simplify, not an answer.
- Which derivative rules does this chapter give?
- The power rule (x^n becomes n x^(n-1)), the derivatives of sin x and cos x, and the algebra rules - sums differentiate term by term, products by the product rule, quotients by the quotient rule.
- Are these the NCERT exercise questions for Limits and Derivatives (Class 11)?
- No. EZT pages are original practice following the chapter's section order - use them to prepare for the textbook exercises, not as a substitute for your book.