Limits – Left & Right Hand Limits
limx→a f(x) exists ⇔ LHL = RHL = finite value
LHL = limx→a⁻ f(x), RHL = limx→a⁺ f(x)
LHL = limx→a⁻ f(x), RHL = limx→a⁺ f(x)
Indeterminate Forms
- 0/0, ∞/∞, 0×∞, ∞−∞, 1^∞, 0^0, ∞^0
- Use L'Hôpital's rule, series, or rationalisation
Standard Limits
Trigonometric
limx→0 sinx/x = 1
limx→0 tanx/x = 1
limx→0 (1−cosx)/x² = 1/2
limx→0 tanx/x = 1
limx→0 (1−cosx)/x² = 1/2
Exponential & Log
limx→0 (1+x)1/x = e
limx→∞ (1+1/x)x = e
limx→0 (e^x − 1)/x = 1
limx→0 ln(1+x)/x = 1
limx→∞ (1+1/x)x = e
limx→0 (e^x − 1)/x = 1
limx→0 ln(1+x)/x = 1
Other Important
limx→a (xⁿ − aⁿ)/(x − a) = naⁿ⁻¹
limx→0 (a^x − 1)/x = ln a
limx→0 (a^x − 1)/x = ln a
Taylor & Maclaurin Series Expansions
sin x
x − x³/3! + x⁵/5! − x⁷/7! + ...
cos x
1 − x²/2! + x⁴/4! − x⁶/6! + ...
ex
1 + x + x²/2! + x³/3! + ...
ln(1+x)
x − x²/2 + x³/3 − x⁴/4 + ... (|x|<1)
(1+x)n
1 + nx + n(n−1)x²/2! + ...
Continuity
f(x) is continuous at x = a if limx→a f(x) = f(a)
(i.e. LHL = RHL = f(a))
(i.e. LHL = RHL = f(a))
Types of Discontinuity
- Removable (limit exists, f(a) ≠ limit or undefined)
- Jump (LHL ≠ RHL)
- Infinite (limit → ±∞)
- Oscillating (sin(1/x) type)
Differentiability
f'(a) = limh→0 [f(a+h) − f(a)] / h
= LHD = RHD at x = a
= LHD = RHD at x = a
Differentiability ⇒ Continuity
Continuity ⇏ Differentiability (e.g. |x| at x=0)
Standard Derivatives
- (xⁿ)' = nxⁿ⁻¹
- (sin x)' = cos x, (cos x)' = −sin x
- (ex)' = ex, (ln x)' = 1/x
- Chain rule, product rule, quotient rule
Rolle’s & Lagrange's Mean Value Theorem
Rolle’s Theorem
If f(a) = f(b), f continuous on [a,b], differentiable on (a,b) ⇒ ∃ c ∈ (a,b) s.t. f'(c) = 0
LMVT
f continuous on [a,b], differentiable on (a,b) ⇒ ∃ c ∈ (a,b) s.t. f'(c) = [f(b)−f(a)]/(b−a)
JEE Main Tips & Weightage
Highest weightage chapter — 6–8 questions every year!
Must Master
- All standard limits & series expansions
- L'Hôpital's rule applications
- lim (sin ax)/x, (1+x)1/x, etc.
- Continuity in pieces & |x| type functions
- Differentiability of composite functions
- Rolle’s/LMVT applications (equating derivatives)
- Sandwich theorem