Limits and Derivatives — Class 11 CBSE Mathematics revision notes
A 4-page visual revision sheet for Limits and Derivatives, written for Class 11 CBSE Mathematics. Formulas, diagrams and the traps that cost marks, laid out to be read the night before rather than the term before. Free, and it opens inside CapCompass.
Page 1 as PDF
Limits and Derivatives in one look
what a function does NEAR a point, and the slope that idea builds
What a limit means
- lim_xto af(x)=L means f(x) gets arbitrarily close to L as x gets close to a, from either side. f itself need not even be defined AT a
- left-hand limit: lim_xto a^-f(x), x approaches a from values LESS than a
- right-hand limit: lim_xto a^+f(x), x approaches a from values GREATER than a
- the two-sided limit exists exactly when the left-hand and right-hand limits both exist AND are equal
Algebra of limits
- sum/difference
lim_xto a[f(x)± g(x)]=lim_xto af(x)±lim_xto ag(x) - product
lim_xto a[f(x)g(x)]=lim_xto af(x)×lim_xto ag(x) - quotient, provided lim_xto ag(x)≠0
lim_xto af(x)/g(x)=lim_xto af(x)/lim_xto ag(x) - constant multiple
lim_xto a[kf(x)]=klim_xto af(x)
Worked: a limit by factoring
Q. evaluate lim_xto2x^2-4/x-2
- direct substitution gives 0/0, undefined, so substitution alone won't work
- factor the numerator: x^2-4=(x-2)(x+2)
- cancel the common factor, valid since x≠2 as the limit is approached, never reached: (x-2)(x+2)/x-2=x+2
- now substitute directly: lim_xto2(x+2)=2+2=4
Answer: the limit is 4
Limits of trigonometric functions
- the key one, x in radians
lim_xto0sin x/x=1 - follows from the one above
lim_xto0tan x/x=1 - a related limit
lim_xto01-cos x/x=0 - a different power of x
lim_xto01-cos x/x^2=1/2
Page 1 of 4. The full sheet, all 4 pages, opens free in the app.
What this Limits and Derivatives sheet covers
- What a limit means
- Algebra of limits
- sum/difference
- product
- quotient, provided lim_xto ag(x)≠0
- constant multiple
- Worked: a limit by factoring
- Limits of trigonometric functions
- the key one, x in radians
- follows from the one above
- a related limit
- a different power of x
- Evaluating a limit: which technique
- Approaching a ceiling
- The derivative, defined as a limit
- The tangent as a limit
- Worked: derivative of x² from first principles
- Rules of differentiation
- a constant
- power rule
- Product and quotient rules
- product rule
- quotient rule, g(x)≠0
- Derivatives worth knowing by heart
- sin x
- cos x
- tan x
- a constant function
- Symbols
- Traps that cost marks
- Tricks that save time
How CapCompass uses this sheet
These notes are not a standalone PDF. CapCompass plans Mathematics chapter by chapter against your real exam dates and free hours, and when the plan reaches Limits and Derivatives the sheet opens inside the study block itself — at the page covering the sub-topic that block is on, with the timer still running. Revision arrives at the moment the plan says revise, instead of having to be gone looking for.