Limits and Derivatives

$lim_{x→a} f(x) = L means f(x) → L as x → a$

The limit L exists if the $left-hand$ and $right-hand$ limits both equal L.

$If \lim f(x)=L and \lim g(x)=M, then \lim [f±g]=L±M, \lim [f·g]=L·M, \lim [\frac{f}{g}]=\frac{L}{M} (M≠0)$

Limits of sums, differences, products, quotients and scalar multiples combine $term-by-term,$ provided each individual limit exists.

$If g(x) ≤ f(x) ≤ h(x) near a, and \lim g(x) = \lim h(x) = L, then \lim f(x) = L$

Used to prove standard limits like $sin \frac{x}{x} → 1$ by trapping f(x) between two simpler functions with the same limit.

$lim_{x→0} (sin x)/x = 1 (x in radians)$

Fundamental trigonometric limit.

$lim_{x→0} (tan x)/x = 1$

$lim_{x→0} (1 − cos x)/x² = \frac{1}{2}$

$lim_{x→0} (eˣ − 1)/x = 1$

Equivalent to saying the derivative of eˣ at $x=0$ is $1.$

$lim_{x→0} ln(1+x)/x = 1$

$lim_{x→a} (xⁿ − aⁿ)/(x − a) = n·aⁿ⁻¹$

Generalises the $difference-of-powers$ factoring trick; used directly to derive the power rule from first principles.

$f'(x) = lim_{h→0} [f(x+h) − f(x)] / h$

The instantaneous rate of change of f at x.

$d/dx (xⁿ) = nxⁿ⁻¹$

$(uv)' = u'v + uv'$

$(\frac{u}{v})' = (u'v − uv') / v²$

$d/dx f(g(x)) = f'(g(x)) · g'(x)$

Used to differentiate composite functions: 'derivative of outer (at inner) times derivative of inner'.

$d/dx(sin x)=cos x; d/dx(cos x)=−sin x; d/dx(tan x)=sec²x; d/dx(eˣ)=eˣ; d/dx(ln x)=\frac{1}{x}$

$If \lim f(x)/g(x) is \frac{0}{0} or ∞/∞, then \lim f(x)/g(x) = \lim f'(x)/g'(x) (if the latter exists)$

A calculus shortcut for indeterminate forms — useful for verifying answers, but NOT part of the NCERT Class $11$ syllabus and should not replace $first-principles/standard-limit$ methods in board answers.