Limits and Continuity — JEE Mains Mathematics

Formulas (2)

• The standard limits · Standard limits
  $$\lim_{x\to0}\frac{\sin x}x=1,\quad\lim_{x\to0}\frac{1-\cos x}{x^2}=\frac12,\quad\lim_{x\to0}\frac{e^x-1}x=1$$
• Factorise or rationalise · Conjugate
  $$\sqrt p-\sqrt q=\frac{p-q}{\sqrt p+\sqrt q}$$

Watch out for (2)

• Match the argument exactly
  → The standard limits
• Rationalise the right part
  → Factorise or rationalise

Formulas (2)

• Limits by series expansion · Key expansions
  $$\sin x=x-\tfrac{x^3}{6}+\cdots,\quad\cos x=1-\tfrac{x^2}{2}+\cdots,\quad e^x=1+x+\tfrac{x^2}{2}+\cdots$$
• Constants that make a limit finite · Matching coefficients
  $$f(x)=c_0+c_1x+\dots:\ \lim_{x\to0}\frac{f(x)}{x^n}\ \text{finite}\Rightarrow c_0=\dots=c_{n-1}=0$$

Watch out for (2)

• Expand far enough
  → Limits by series expansion
• Finite is not the same as zero
  → Constants that make a limit finite

Formulas (2)

• The 1^∞ formula at a point · 1 to the power infinity
  $$\lim f(x)^{g(x)}=e^{\lim g(x)\,(f(x)-1)}\quad(f\to1,\ g\to\infty)$$
• The 1^∞ formula at infinity · The e limit
  $$\lim_{n\to\infty}\left(1+\frac an\right)^{bn}=e^{ab}$$

Watch out for (2)

• Is it really 1 to the infinity?
  → The 1^∞ formula at a point
• The product can tend to 0
  → The 1^∞ formula at infinity

Formulas (3)

• The dominant terms · Leading terms
  $$\lim_{x\to\infty}\frac{a_nx^n+\dots}{b_nx^n+\dots}=\frac{a_n}{b_n}$$
• Sum first, then take the limit · Power sums
  $$\sum_{k=1}^nk^2=\frac{n(n+1)(2n+1)}6$$
• Sums as integrals · Riemann sum
  $$\lim_{n\to\infty}\frac1n\sum_{k=1}^{n}f\!\left(\frac kn\right)=\int_0^1f(x)\,dx$$

Watch out for (3)

• Infinity minus infinity is not 0
  → The dominant terms
• Many small terms can add to something
  → Sum first, then take the limit
• Find the right limits of integration
  → Sums as integrals

Formulas (2)

• Limits that are derivatives · Derivative as a limit
  $$f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}$$
• Limits of integrals · Leibniz rule
  $$\frac{d}{dx}\int_a^{u(x)}g(t)\,dt=g\big(u(x)\big)\,u'(x)$$

Watch out for (2)

• Check the form before L'Hospital
  → Limits that are derivatives
• Chain rule on the limit
  → Limits of integrals

Formulas (2)

• The greatest integer function · Greatest integer bounds
  $$t-1<[t]\le t$$
• Left and right limits · Existence of a limit
  $$\lim_{x\to a}f\ \text{exists}\ \Leftarrow\ \lim_{x\to a^-}f=\lim_{x\to a^+}f$$

Watch out for (2)

• Values near an integer from above
  → The greatest integer function
• sin|x| over x
  → Left and right limits

Formulas (2)

• Matching the two sides · Continuity
  $$\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=f(a)$$
• Using continuity · Intermediate value theorem
  $$f(a)\,f(b)<0\ \Rightarrow\ f(c)=0\ \text{for some}\ c\in(a,b)$$

Watch out for (2)

• Use the right formula on each side
  → Matching the two sides
• No sign change, no guarantee
  → Using continuity

Formulas (3)

• Jumps and corners · Where the greatest integer jumps
  $$[g(x)]\ \text{can break only where}\ g(x)\in\mathbb Z$$
• Compositions · Candidate points
  $$\{x:g\ \text{breaks}\}\cup\{x:g(x)\ \text{is a break of}\ f\}$$
• Functions defined as a limit · Powers in the limit
  $$\lim_{n\to\infty}x^{2n}=\begin{cases}0,&|x|<1\\1,&|x|=1\\\infty,&|x|>1\end{cases}$$

Watch out for (3)

• A zero factor can hide a jump
  → Jumps and corners
• Also where the inside lands on a bad point
  → Compositions
• Compute the boundary separately
  → Functions defined as a limit