Formula Sheet — Limits and Derivatives

Card 1: Limits

Fact Statement
Existence limit exists ⇔\Leftrightarrow LHL == RHL; that common value is the limit
Limit vs value lim⁡x→af(x)\lim_{x \to a} f(x) and f(a)f(a) are independent — either can exist alone
Limit laws lim⁡[f±g]=lim⁡f±lim⁡g\lim[f \pm g] = \lim f \pm \lim g; lim⁡[fg]=lim⁡f⋅lim⁡g\lim[fg] = \lim f \cdot \lim g; lim⁡fg=lim⁡flim⁡g\lim\frac{f}{g} = \frac{\lim f}{\lim g} (lim⁡g≠0\lim g \ne 0)
Polynomials lim⁡x→af(x)=f(a)\lim_{x \to a} f(x) = f(a) — direct substitution
00\frac{0}{0} rational factor out (x−a)(x - a), cancel, substitute again
Sandwich f≤g≤hf \leq g \leq h, lim⁡f=lim⁡h=l\lim f = \lim h = l ⇒\Rightarrow lim⁡g=l\lim g = l

Card 2: Standard limits

Limit Value
lim⁡x→axn−anx−a\lim_{x \to a}\frac{x^n - a^n}{x - a} nan−1na^{n-1} (any rational nn)
lim⁡x→0sin⁡xx\lim_{x \to 0}\frac{\sin x}{x} 11 (radians)
lim⁡x→01−cos⁡xx\lim_{x \to 0}\frac{1 - \cos x}{x} 00
lim⁡x→0tan⁡xx\lim_{x \to 0}\frac{\tan x}{x} 11
lim⁡x→0ex−1x\lim_{x \to 0}\frac{e^x - 1}{x} (JEE) 11
lim⁡x→0ax−1x\lim_{x \to 0}\frac{a^x - 1}{x} (JEE) ln⁡a\ln a
lim⁡x→0log⁡(1+x)x\lim_{x \to 0}\frac{\log(1+x)}{x} (JEE) 11
lim⁡x→0(1+x)1/x\lim_{x \to 0}(1 + x)^{1/x} (JEE) ee

Card 3: Derivatives

Tool Statement
At a point f′(a)=lim⁡h→0f(a+h)−f(a)hf'(a) = \lim_{h \to 0}\frac{f(a+h) - f(a)}{h} — the tangent slope at aa
First principles f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0}\frac{f(x+h) - f(x)}{h}
Sum / difference (u±v)′=u′±v′(u \pm v)' = u' \pm v'
Product (Leibnitz) (uv)′=u′v+uv′(uv)' = u'v + uv'
Quotient (uv)′=u′v−uv′v2\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}
Power ddxxn=nxn−1\frac{d}{dx}x^n = nx^{n-1} (any real nn)
Trig (sin⁡x)′=cos⁡x(\sin x)' = \cos x; (cos⁡x)′=−sin⁡x(\cos x)' = -\sin x; (tan⁡x)′=sec⁡2x(\tan x)' = \sec^2 x
More trig (cot⁡x)′=−cosec2x(\cot x)' = -\mathrm{cosec}^2 x; (sec⁡x)′=sec⁡xtan⁡x(\sec x)' = \sec x\tan x; (cosec x)′=−cosec xcot⁡x(\mathrm{cosec}\,x)' = -\mathrm{cosec}\,x\cot x

Last-Minute Mistake Checklist

  1. Check LHL and RHL separately for every piecewise or modulus function — one-branch substitution is the classic existence error.
  2. The limit ignores f(a)f(a): never argue "f(1)=0f(1) = 0 so the limit is 0"; the limit reads the neighbourhood, not the point.
  3. Cancel only after confirming 00\frac{0}{0}: if only the denominator vanishes, the limit does not exist — no cancellation can save it.
  4. Radians only in sin⁡xx→1\frac{\sin x}{x} \to 1; a degree version picks up π180\frac{\pi}{180}.
  5. Match arguments before using sin⁡θθ→1\frac{\sin\theta}{\theta} \to 1: sin⁡4xx→4\frac{\sin 4x}{x} \to 4, not 1 — multiply and divide by the argument first.
  6. Product rule is u′v+uv′u'v + uv', never u′v′u'v'; quotient rule numerator starts with u′vu'v, and the sign between the terms is minus.
  7. "From first principles" is binding: write the h→0h \to 0 quotient even when the table answer is obvious.
  8. ddxcos⁡x=−sin⁡x\frac{d}{dx}\cos x = -\sin x — the cosine sign flip is the single most common derivative slip.
  9. Constants die, coefficients survive: (5sin⁡x+7)′=5cos⁡x(5\sin x + 7)' = 5\cos x; the 7 vanishes, the 5 does not.
  10. Differentiable ⇒\Rightarrow continuous, never conversely: ∣x∣|x| at 0 is the standing counterexample — corners kill derivatives, not continuity.

How this chapter flows on: Class 12 begins with continuity and differentiability, then applications of derivatives (tangents, maxima-minima, rates) and integral calculus — every one of them a direct continuation of these limit and derivative definitions.