CBSE Class 11 · Mathematics
Limits and Derivatives
Official NCERT chapter from Mathematics (book code kemh1). ExamMaster notes are original teaching at CBSE Class 11 depth.
This lesson follows the official chapter “Limits and Derivatives” in Mathematics. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.
- CBSE Class 11
- Medium level
- 8 concepts
1Intuitive Idea of Derivatives
A derivative, intuitively, is a slope the graph is settling to: a secant’s rise/run as the run shrinks. For y=x² near x=3, the chord from 3 to 3.1 has slope 6.1; closer runs hug 6. The idea is that hug, not a vertical jump.
A single fat chord is not yet the derivative.
Figure. The derivative at P is the tangent slope, not the dashed chord from the origin.
How it works
- Pick a nearby second pointThe secant.
- Shrink the runThe hug.
- Name the settled slope as the derivative at that xThe idea.
2Limits
A limit is the value a writing approaches as x approaches a number (or infinity, if taught). lim(x→2) (x²−4)/(x−2) = 4 because the hole at 2 is removable as (x+2). A limit is a settle, not “plug if it blows up and shrug”.
Declaring “does not exist” because you saw /0 before simplifying is a rush.
Figure. Both sides run toward 2. The hole means the limit need not equal a written f(1).
How it works
- See what x is approachingThe approach.
- Simplify if a hole is removable, as taughtThe prepare.
- Read the settle — or say it does not settle if left and right disagreeThe limit.
(x²−4)/(x−2) as x→2
Find lim(x→2) (x²−4)/(x−2).
- Factor(x−2)(x+2)/(x−2)
- For x≠2x+2
- Limit4
Pro tip. A hole you can cancel is a settle, not a shrug.
3Limits of Trigonometric Functions
Trig limits at this class include the taught spine lim(θ→0) sin θ / θ = 1 (θ in radians). A degree-mode θ is a different number-story. Use it to finish writings like sin(3x)/(5x) → 3/5 as x→0, as taught.
Plugging θ=0 into sin θ / θ as 0/0 and stopping is the hole-shrug.
Figure. As x goes to 0 the curve meets the dashed height 1: lim (sin x)/x = 1.
How it works
- Name the small-angle writingThe given.
- Rewrite toward sin u / u with u→0The prepare.
- Read 1 times the leftover coefficientThe trig limit.
4Derivatives
A derivative as a limit is lim(h→0) [f(x+h)−f(x)]/h. For f(x)=x² at x=3, the limit is 6. Differentiation rules (power, sum) if taught are shortcuts that must match this limit on a check case.
Calling 2x the derivative of x² without a meaning is a badge.
Figure. The derivative is its own graph. For x squared it is the straight line 2x through the origin.
How it works
- Write the difference quotientThe slope-to-be.
- Let h settle to 0The derivative.
- For x², get 2x; at 3, get 6The match.
5A definition is a test you can run
Limit / derivative is a test: a settle, or a settled slope. If you only say “tends”, you have a heading.
A speedometer photo is not the test.
Figure. Continuity is a three-way test: left limit, right limit, and f(a) must be the same number.
How it works
- Name the writing and what x or h approachesThe object.
- Give the settle or the slopeThe test.
- Then the word has contentThe definition ran.
6Name the given before the unknown
The given is f and the point. The unknown is the limit or f′. Copy radians before you use a trig limit.
Using h=0 inside the quotient before the limit is a silent /0.
Figure. Name the given polynomial first. Only then ask for the derivative.
How it works
- Copy f and the approachThe given.
- Name the settle or the derivativeThe unknown.
- Then simplify, then settleGiven first.
7One worked case is enough at this class
One removable hole or one power-check is enough. A page of the same (x²−4)/(x−2) clone does not add a new sin θ / θ.
Ten twins of 4 are still one idea.
Figure. One worked derivative is term by term: 4x^3 to 12x^2, -9x to -9, constant 2 to 0.
How it works
- Run one limit or one difference-quotientThe case.
- A second only as a checkOptional.
- StopThis class.
8A miss: swapping the school name for the picture
A speedometer photo is a setting. The maths is a difference quotient. A needle is not f′(3).
A racecar still is not lim sin θ / θ.
Figure. This chapter’s picture is a tangent on a curve. An AP staircase belongs to a different chapter.
How it works
- Keep the dash as a storyA setting.
- Write [f(x+h)−f(x)]/hThe maths.
- Then settle h→0Do not swap.
lim(x→2) (x²−4)/(x−2) is
- 4
- 0
- Does not exist because of /0
Cancel the hole; settle at 4.
Notes
- The official chapter title is “Limits and Derivatives”. Teach the school test for that title, not a contest shortcut.
- If a step needs a later class, stop. The next official chapter will pick it up.
Formulas
- f′(x)=lim(h→0)[f(x+h)−f(x)]/h
- lim(θ→0) sinθ/θ = 1 (radians)
Recap
Hold these pegs from the official chapter “Limits and Derivatives”. The wording is ExamMaster’s teaching, not a textbook recap.
- Intuitive Idea of Derivatives
- A derivative, intuitively, is a slope the graph is settling to: a secant’s rise/run as the run shrinks.
- Limits
- A limit is the value a writing approaches as x approaches a number (or infinity, if taught).
- Limits of Trigonometric Functions
- Trig limits at this class include the taught spine lim(θ→0) sin θ / θ = 1 (θ in radians).
- Derivatives
- A derivative as a limit is lim(h→0) [f(x+h)−f(x)]/h.
- A definition is a test you can run
- Limit / derivative is a test: a settle, or a settled slope.
- Name the given before the unknown
- The given is f and the point. The unknown is the limit or f′. Copy radians before you use a trig limit.
Practise Limits and Derivatives
Reading is free and needs no account. Practice, mocks and progress live in the app.
- A 4-question practice set that ends the chapter
- 1 quick check with worked explanations
- Timed mocks scored with the real marking scheme
- Readiness tracked per topic, kept on your device