Limits — Class 11 Maharashtra HSC Mathematics revision notes
A 4-page visual revision sheet for Limits, written for Class 11 Maharashtra HSC 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 in one look
what f(x) is heading toward as x closes in on a, from both sides at once
What a limit means
- xto a means x takes values closer and closer to a, but never equal to a
- if a polynomial P(x) is evaluated, lim_xto aP(x)=P(a) directly: no special technique needed
One-sided limits
- notation: lim_xto a^-f(x)
- x approaches a through values LESS than a
- reads off behaviour just before a
- notation: lim_xto a^+f(x)
- x approaches a through values GREATER than a
- reads off behaviour just after a
Algebra of limits
- shorthand for this whole block
L=lim_xto af(x), M=lim_xto ag(x) - sum, difference
lim_xto a[f(x)± g(x)]=L± M - product
lim_xto a[f(x)g(x)]=LM - quotient, provided M≠0
lim_xto af(x)/g(x)=L/M - constant multiple
lim_xto a[kf(x)]=kL
Quotient and infinity limits, the special cases
- lim g(x)≠0 — quotient rule applies directly
- lim g(x)=0, lim f(x)≠0 — the limit does not exist
- lim g(x)=0=lim f(x) — 0/0 form: factor, rationalize, or use a standard limit
- xto∞, equal degrees top and bottom — limit is the ratio of leading coefficients
- xto∞, numerator degree smaller — limit is 0
- xto∞, numerator degree larger — limit does not exist, grows unbounded
Worked: a 0/0 limit by factoring
Q. evaluate lim_xto2x^2-4/x-2
- direct substitution gives 0/0, substitution alone will not settle it
- factor the numerator: x^2-4=(x-2)(x+2)
- cancel the common factor, valid since x≠2: (x-2)(x+2)/x-2=x+2
- now substitute directly: 2+2
Answer: the limit is 4
Page 1 of 4. The full sheet, all 4 pages, opens free in the app.
What this Limits sheet covers
- What a limit means
- One-sided limits
- left-hand, x to a from below
- right-hand, x to a from above
- Algebra of limits
- shorthand for this whole block
- sum, difference
- product
- quotient, provided M≠0
- constant multiple
- Quotient and infinity limits, the special cases
- Worked: a 0/0 limit by factoring
- The standard power theorem
- for n a positive integer, a>0
- Worked: applying the power theorem
- Evaluating P(x) over Q(x) as x approaches a: which case
- Worked: a limit by rationalizing
- Limits of trigonometric functions
- the key one, θ in radians
- follows from the one above
- scaled by a constant p inside
- sin, cos are continuous
- Squeeze (sandwich) theorem
- Worked: the squeeze theorem
- Exponential and logarithmic limits
- the natural base
- any base a>0
- the natural log
- the defining limit for e
- Worked: an exponential-difference limit
- Limits at infinity
- Worked: a limit at infinity
- Infinite limits and vertical asymptotes
- 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 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.
More Mathematics Class 11 revision notes
- Angle and its Measurement
- Circle
- Complex Numbers
- Conic Sections
- Continuity
- Determinants and Matrices
- Differentiation
- Functions
- Measures of Dispersion
- Methods of Induction and Binomial Theorem
- Permutations and Combination
- Probability
- Sequences and Series
- Sets and Relations
- Straight Line
- Trigonometry - 1
- Trigonometry - 2