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Limits revision notes

Every idea in the Limits chapter for JEE Main, with the usual mistakes and the fast methods. Topic 1 is open to everyone; sign in free to read all 22 and download the PDF.

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How we count: 7× asked means 7 past JEE Main questions in the PrepWiser bank needed that concept, and "last" is the latest year one appeared. ★ marks ideas asked 8 or more times.

Limit and one-sided limits

Left and right limits
6× asked, last 2024
lim⁡x→af(x)=L  ⟺  lim⁡x→a−f(x)=lim⁡x→a+f(x)=L\lim_{x\to a}f(x)=L\iff\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=L
LL finite; f(a)f(a) need not exist or equal LL. If a parameter must make the limit exist, write LHL and RHL in it and equate.
xxyyRHL 11LHL −1-1
∣x∣x\frac{|x|}x: two different one-sided limits
xxyy→1\to1→−1\to-1
e1/x−1e1/x+1\frac{e^{1/x}-1}{e^{1/x}+1}: e1/x→0e^{1/x}\to0 from the left, ∞\infty from the right
  • Not ∣x∣/x→1|x|/x\to1. It is −1-1 on the left and +1+1 on the right, so a∣x∣/xa|x|/x has two one-sided limits. Equate them to find aa; sin⁡∣x∣\sin|x| over xx flips sign the same way.
  • Not 1−cos⁡θ≈θ/2\sqrt{1-\cos\theta}\approx\theta/\sqrt2. A root is never negative, so it is ∣θ∣/2|\theta|/\sqrt2. Fix the sign of θ\theta from the side of approach; ∣sin⁡x∣|\sin x| works the same way.
  • Not one value for e1/xe^{1/x} or tan⁡−1(1/x)\tan^{-1}(1/x) at 0. 1/x→−∞1/x\to-\infty on the left, +∞+\infty on the right. So e1/x→0e^{1/x}\to0 or ∞\infty, and tan⁡−1(1/x)→−π/2\tan^{-1}(1/x)\to-\pi/2 or π/2\pi/2. Ex: (e1/x−1)/(e1/x+1)→−1(e^{1/x}-1)/(e^{1/x}+1)\to-1 or 11.
Greatest integer and fractional part
8× asked, last 2024
lim⁡x→n−[x]=n−1,lim⁡x→n+[x]=n,lim⁡x→n−{x}=1,lim⁡x→n+{x}=0\lim_{x\to n^-}[x]=n-1,\quad \lim_{x\to n^+}[x]=n,\qquad \lim_{x\to n^-}\{x\}=1,\quad \lim_{x\to n^+}\{x\}=0
nn an integer, {x}=x−[x]\{x\}=x-[x]. Both jump at every integer and are continuous elsewhere. [x+n]=[x]+n[x+n]=[x]+n.
xxyy12312x→2−x\to2^-: 1x→2+x\to2^+: 2
[x][x] jumps at every integer: LHL n−1n-1, RHL nn
xxyy1231
{x}\{x\}: LHL 1, RHL 0 at each integer
  • [g(x)][g(x)] as x→ax\to a with g(x)→ℓg(x)\to\ell: if ℓ\ell is not an integer, the limit is [ℓ][\ell]. If ℓ=k\ell=k is an integer, it is kk when g→kg\to k from above and k−1k-1 from below; find that side first.
  • {x}\{x\} at 0: on the right {x}=x→0+\{x\}=x\to0^+, on the left {x}=1+x→1−\{x\}=1+x\to1^-. So cos⁡−1(1−{x}2)\cos^{-1}(1-\{x\}^2) behaves like 2 x\sqrt2\,x on one side and tends to π/2\pi/2 on the other.
  • Not [−x]=−[x][-x]=-[x]. As x→7+x\to7^+, 1−x→−6−1-x\to-6^- so [1−x]=−7[1-x]=-7; as x→7−x\to7^- it is −6-6. Find which way the inside moves before applying [ ][\,].
Algebra of limits and substitution
4× asked, last 2025
lim⁡(f±g)=L±M,lim⁡fg=LM,lim⁡fg=LM (M≠0),lim⁡g(f(x))=g(L)\lim(f\pm g)=L\pm M,\quad \lim fg=LM,\quad \lim\frac fg=\frac LM\ (M\ne0),\qquad \lim g(f(x))=g(L)
L,ML,M finite; the last needs gg continuous at LL. Where the expression is continuous, substitute directly.

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Practise Limits

19 more concepts in 5 topics

Algebraic methods for 0/0 and ∞/∞; Standard limits and the form 1 to the power ∞; Series expansions and unknown constants; L'Hospital's rule and derivative limits; and Sandwich theorem, sums and sequences. Read the first part free. Sign in free with Google to read the rest and download the PDF.

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Algebraic methods for 0/0 and ∞/∞

  • Indeterminate forms
  • Factorisation and the xⁿ − aⁿ limit6×
  • Rationalisation9×
  • Limits at infinity: highest power5×

Standard limits and the form 1 to the power ∞

  • Trigonometric standard limits16×
  • Exponential and logarithmic limits9×
  • e as a limit2×
  • One to the power infinity14×

Series expansions and unknown constants

  • Exponential and log series13×
  • Binomial series4×
  • Trigonometric and inverse-trig series11×
  • Unknown constants for a finite limit19×

L'Hospital's rule and derivative limits

  • L'Hospital's rule5×
  • Limits with integrals11×
  • Derivative as a limit5×

Sandwich theorem, sums and sequences

  • Sandwich theorem5×
  • x[k/x] near 01×
  • Limits of sums (Riemann sums)3×
  • Sequences: powers and nth roots1×

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