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Limits formula sheet

Every formula in the Limits chapter for JEE Main, with one line on when it applies. 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 formula, and "last" is the latest year one appeared. ★ marks formulas 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=lim⁡x→a+f=L\lim_{x\to a}f(x)=L\iff\lim_{x\to a^-}f=\lim_{x\to a^+}f=L
∣x∣/x→∓1|x|/x\to\mp1; 1−cos⁡θ≈∣θ∣/2\sqrt{1-\cos\theta}\approx|\theta|/\sqrt2. At 0: e1/x→0e^{1/x}\to0 or ∞\infty; tan⁡−1(1/x)→∓π/2\tan^{-1}(1/x)\to\mp\pi/2.
[x] and {x}
8× asked, last 2024
lim⁡x→n−[x]=n−1,lim⁡x→n+[x]=nlim⁡x→n−{x}=1,lim⁡x→n+{x}=0\begin{aligned}&\lim_{x\to n^-}[x]=n-1,\quad \lim_{x\to n^+}[x]=n\\&\lim_{x\to n^-}\{x\}=1,\quad \lim_{x\to n^+}\{x\}=0\end{aligned}
[g(x)][g(x)], g→kg\to k integer: kk from above, k−1k-1 from below. Find the side of the inside first.
Algebra of limits
4× asked, last 2025
lim⁡(f±g)=L±M,lim⁡fg=LMlim⁡fg=LM (M≠0)\begin{aligned}&\lim(f\pm g)=L\pm M,\qquad \lim fg=LM\\&\lim\frac fg=\frac LM\ (M\ne0)\end{aligned}
lim⁡g(f(x))=g(L)\lim g(f(x))=g(L) for continuous gg. Where continuous, substitute.
Indeterminate forms
00, ∞∞, ∞−∞, 0⋅∞, 1∞, 00, ∞0\frac00,\ \frac\infty\infty,\ \infty-\infty,\ 0\cdot\infty,\ 1^\infty,\ 0^0,\ \infty^0
Last three: fg=egln⁡ff^g=e^{g\ln f}. 0∞0^\infty is not indeterminate.

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Practise Limits

18 more formulas in 5 topics

Algebraic methods; Standard limits and 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

  • xⁿ − aⁿ6×
  • Rationalisation9×
  • Highest power5×

Standard limits and 1 to the power ∞

  • Trigonometric16×
  • Exponential and log9×
  • e as a limit2×
  • 1 to the power ∞14×

Series expansions and unknown constants

  • Exponential and log series13×
  • Binomial series4×
  • Trigonometric series11×
  • Unknown constants19×

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×
  • Riemann sums3×
  • Powers and nth roots1×

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About 110 past JEE Main questions from this chapter, with instant answer checking and step-by-step solutions where available. Free, and you can start without signing up.

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