Limits and Continuity — MHT-CET Maths

Formulas (6)

• What a Limit Says · Algebra of limits
  $$\lim (f \pm g) = \lim f \pm \lim g \qquad \lim (fg) = \lim f \cdot \lim g \qquad \lim \frac{f}{g} = \frac{\lim f}{\lim g}\ \ (\lim g \neq 0)$$
• One-Sided Limits and When a Limit Exists · Existence of a limit
  $$\lim_{x\to a} f(x) = L \iff \lim_{x\to a^-} f(x) = \lim_{x\to a^+} f(x) = L$$
• Greatest-Integer and Sign Functions Near a Point · Greatest integer at an integer
  $$\lim_{x\to n^-}[x] = n - 1, \qquad \lim_{x\to n^+}[x] = n \qquad (n \in \mathbb{Z})$$
• Limits at Infinity — Compare the Leading Powers · Ratio of polynomials at infinity
  $$\lim_{x\to\infty}\frac{a_n x^n + \dots + a_0}{b_m x^m + \dots + b_0} = \begin{cases} 0, & n < m \\[2pt] \dfrac{a_n}{b_m}, & n = m \\[2pt] \pm\infty, & n > m \end{cases}$$
• A Finite Limit at Infinity Forces the Divergent Part to Vanish · Finite limit at infinity
  $$\frac{p(x)}{q(x)} = \alpha x + \beta + \frac{r(x)}{q(x)} \ \Rightarrow\ \lim_{x\to\infty}\left(\frac{p(x)}{q(x)} - ax - b\right) \text{ finite} \iff a = \alpha,\ \text{and the limit is } \beta - b$$
• Infinity Minus Infinity — Rationalise at Infinity · Root minus its leading term
  $$\lim_{x\to\infty}\left(\sqrt{x^2 + ax + b} - x\right) = \frac{a}{2}$$

Watch out for (6)

• The value at the point is not the limit
  → What a Limit Says
• Not every modulus makes the limit fail
  → One-Sided Limits and When a Limit Exists
• [x] for a small negative x is −1, not 0
  → Greatest-Integer and Sign Functions Near a Point
• A sum of n terms is not 'n copies of the biggest term'
  → Limits at Infinity — Compare the Leading Powers
• Solving for b before a
  → A Finite Limit at Infinity Forces the Divergent Part to Vanish
• Subtracting infinities term by term
  → Infinity Minus Infinity — Rationalise at Infinity

Formulas (5)

• Factor and Cancel · Factorisations that unlock 0/0
  $$x^2 - a^2 = (x - a)(x + a) \qquad x^3 - a^3 = (x - a)(x^2 + ax + a^2) \qquad p(a) = 0 \Rightarrow (x - a) \mid p(x)$$
• Rationalisation — Single, Double and Nested Surds · Conjugate rule
  $$\frac{\sqrt{A} - \sqrt{B}}{C} = \frac{A - B}{C\left(\sqrt{A} + \sqrt{B}\right)}$$
• The xⁿ − aⁿ Standard Form and Fractional Powers · The xⁿ − aⁿ family
  $$\lim_{x\to a}\frac{x^n - a^n}{x - a} = n a^{n-1} \quad (n \in \mathbb{Q}) \qquad \lim_{x\to 0}\frac{(1 + x)^n - 1}{x} = n$$
• The Derivative in Disguise — [f(x) − f(a)]/(x − a) and L'Hôpital · Derivative form and L'Hôpital
  $$\lim_{x\to a}\frac{f(x) - f(a)}{x - a} = f'(a) \qquad \lim_{x\to a}\frac{f(x)}{g(x)} \overset{0/0}{=} \lim_{x\to a}\frac{f'(x)}{g'(x)}$$
• A Finite Limit Forces the Numerator to Vanish — Finding a and b · Finite limit at a zero of the denominator
  $$\lim_{x\to a}\frac{p(x)}{x - a} = L \text{ (finite)} \ \Rightarrow\ p(a) = 0 \ \text{ and } \ p'(a) = L$$

Watch out for (6)

• Cancelling before checking the form
  → Spotting the 0/0 Form
• Two blowing-up fractions must be combined first
  → Factor and Cancel
• Rationalising only one floor when both carry surds
  → Rationalisation — Single, Double and Nested Surds
• The inner function's sign
  → The xⁿ − aⁿ Standard Form and Fractional Powers
• L'Hôpital on a form that is not indeterminate
  → The Derivative in Disguise — [f(x) − f(a)]/(x − a) and L'Hôpital
• Treating a as free and reading b off the limit
  → A Finite Limit Forces the Numerator to Vanish — Finding a and b

Formulas (6)

• sin x/x, tan x/x and Scaled Arguments · The sine and tangent standard limits
  $$\lim_{x\to 0}\frac{\sin x}{x} = 1 \qquad \lim_{x\to 0}\frac{\tan x}{x} = 1 \qquad \lim_{x\to 0}\frac{\sin kx}{x} = k \qquad \lim_{x\to 0}\frac{\sin ax}{\sin bx} = \frac{a}{b}$$
• The 1 − cos x Family: (1 − cos kx)/x² = k²/2 · 1 − cos x and its scaling
  $$1 - \cos x = 2\sin^2\frac{x}{2} \qquad \lim_{x\to 0}\frac{1 - \cos x}{x^2} = \frac{1}{2} \qquad \lim_{x\to 0}\frac{1 - \cos kx}{x^2} = \frac{k^2}{2}$$
• Rewrite with an Identity Before Taking the Limit · Identities that expose a standard form
  $$\sin(\pi - \theta) = \sin\theta \qquad \cos A - \cos B = -2\sin\frac{A + B}{2}\sin\frac{A - B}{2} \qquad \sqrt{2 - 2\cos\phi} = 2\left|\sin\frac{\phi}{2}\right|$$
• Shift the Variable: Limits at π/2 and Other Non-Zero Points · The π/2 shift
  $$x = \tfrac{\pi}{2} - h:\quad \sin x = \cos h,\ \ \cos x = \sin h,\ \ \cot x = \tan h,\ \ \pi - 2x = 2h,\ \ 1 - \sin x = 1 - \cos h \sim \tfrac{h^2}{2}$$
• Degrees Are Not Radians · Degree conversion in a limit
  $$x^\circ = \frac{\pi x}{180}\ \text{rad} \qquad \lim_{x\to 0}\frac{\sin x^\circ}{x} = \frac{\pi}{180} \qquad \lim_{x\to 0}\frac{1 - \cos x^\circ}{x^2} = \frac{\pi^2}{2\cdot 180^2}$$
• Higher-Order Forms: Expand to the Needed Power · Series to the third order
  $$\sin x = x - \frac{x^3}{6} + \dots \qquad \tan x = x + \frac{x^3}{3} + \dots \qquad \cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \dots \qquad e^x = 1 + x + \frac{x^2}{2} + \dots$$

Watch out for (7)

• sin x/x → 1 only as x → 0
  → sin x/x, tan x/x and Scaled Arguments
• Treating 1 − cos x as first order
  → The 1 − cos x Family: (1 − cos kx)/x² = k²/2
• √(2 − 2cos φ) is 2|sin(φ/2)|, and the modulus decides the sides
  → Rewrite with an Identity Before Taking the Limit
• cos²x does not tend to 0
  → Rewrite with an Identity Before Taking the Limit
• (π − 2x)³ is 8h³, not h³
  → Shift the Variable: Limits at π/2 and Other Non-Zero Points
• Dropping the degree sign
  → Degrees Are Not Radians
• Stopping at first order and getting 0/0 again
  → Higher-Order Forms: Expand to the Needed Power

Formulas (6)

• The Exponential and Logarithmic Standard Limits · Exponential and logarithmic standard limits
  $$\lim_{x\to 0}\frac{a^x - 1}{x} = \log a \qquad \lim_{x\to 0}\frac{e^x - 1}{x} = 1 \qquad \lim_{x\to 0}\frac{\log(1 + x)}{x} = 1 \qquad \lim_{x\to 0}\frac{\log(1 + kx)}{x} = k$$
• Factorising aˣ − bˣ − cˣ + 1 into (bˣ − 1)(cˣ − 1) · Grouping factorisation
  $$(bc)^x - b^x - c^x + 1 = (b^x - 1)(c^x - 1) \sim x^2\,\log b\,\log c$$
• Substitute t = aˣ When the Exponents Are Mixed · Exponential substitution
  $$t = a^{x/k}\ \Rightarrow\ a^{x} = t^{k},\quad a^{c - x} = \frac{a^{c}}{t^{k}},\qquad x \to x_0 \iff t \to a^{x_0/k}$$
• Composite Forms: (eᵘ − 1)/u with u → 0, and Mixed Series Terms · Composite exponential forms
  $$e^{a} - e^{b} = e^{b}\left(e^{a - b} - 1\right) \qquad \lim_{u\to 0}\frac{e^{u} - 1}{u} = 1 \ \text{ for any } u = u(x) \to 0$$
• The 1^∞ Form: lim f^g = e^{lim (f − 1)g} · The 1^∞ rule
  $$\lim_{x\to 0}(1 + x)^{1/x} = e \qquad \lim_{x\to\infty}\left(1 + \frac{k}{x}\right)^{x} = e^{k} \qquad f \to 1,\ g \to \infty:\ \lim f^{g} = e^{\lim (f - 1)g}$$
• 0⁰ and ∞⁰ Forms — Take Logarithms · Power forms through the logarithm
  $$l = \lim f^{g} \ \Rightarrow\ \log l = \lim g\log f \qquad \lim_{x\to 0^+} x\log x = 0 \ \Rightarrow\ \lim_{x\to 0^+} x^{x} = 1$$

Watch out for (6)

• (aˣ − 1)/x is log a, never a
  → The Exponential and Logarithmic Standard Limits
• Sending each term to its own limit
  → Factorising aˣ − bˣ − cˣ + 1 into (bˣ − 1)(cˣ − 1)
• Choosing t too large
  → Substitute t = aˣ When the Exponents Are Mixed
• Using only first order on eˣ² − cos x
  → Composite Forms: (eᵘ − 1)/u with u → 0, and Mixed Series Terms
• 1^∞ is indeterminate — never answer 1 by substitution
  → The 1^∞ Form: lim f^g = e^{lim (f − 1)g}
• Treating (sin x)^(1/x) as a 0⁰ form
  → 0⁰ and ∞⁰ Forms — Take Logarithms

Formulas (6)

• Continuity at a Point — The Three-Part Test · Continuity at c
  $$f \text{ continuous at } c \iff \lim_{x\to c^-} f(x) = \lim_{x\to c^+} f(x) = f(c)$$
• Removable Discontinuity: Define f(c) as the Limit · Filling a removable discontinuity
  $$f \text{ continuous at } c,\ f \text{ undefined at } c \text{ by formula} \ \Rightarrow\ f(c) := \lim_{x\to c} f(x)$$
• Products of Standard Forms in Continuity Dress · Order bookkeeping
  $$\frac{\prod (\text{factors} \sim c_i x^{p_i})}{\prod (\text{factors} \sim d_j x^{q_j})} \to \frac{\prod c_i}{\prod d_j} \ \text{ exactly when } \sum p_i = \sum q_j$$
• Continuity at a Non-Zero Point: Shift to h → 0 · Two routes at a non-zero point
  $$x = c + h \ (h \to 0) \qquad \text{or} \qquad \lim_{x\to c}\frac{p(x)}{q(x)} \overset{0/0}{=} \frac{p'(c)}{q'(c)} \ \text{ when } q'(c) \neq 0$$
• 1^∞ in Continuity Problems: k = e^{…} · The 1^∞ continuity value
  $$f(c) = \lim_{x\to c} u(x)^{v(x)} = e^{\lim_{x\to c}(u - 1)\,v} \qquad (u \to 1,\ v \to \infty)$$
• The Parameter Inside the Function · Parameter through a standard limit
  $$\lim_{x\to 0}\frac{\cos ax - \cos bx}{\cos cx - \cos bx} = \frac{b^2 - a^2}{b^2 - c^2} \qquad \frac{d}{dx}\int_{c}^{u(x)}\phi(t)\,dt = \phi(u(x))\,u'(x)$$

Watch out for (6)

• A limit existing is not continuity
  → Continuity at a Point — The Three-Part Test
• Computing f(c) from the formula
  → Removable Discontinuity: Define f(c) as the Limit
• Powers that do not match
  → Products of Standard Forms in Continuity Dress
• Applying the quotient rule instead of L'Hôpital
  → Continuity at a Non-Zero Point: Shift to h → 0
• A fixed base is not 1^∞
  → 1^∞ in Continuity Problems: k = e^{…}
• Differentiating the integral without the chain factor
  → The Parameter Inside the Function

Formulas (5)

• One Junction: Left Limit = Right Limit = Value · Junction condition
  $$f(x) = \begin{cases} g(x), & x \le c \\ h(x), & x > c \end{cases} \text{ continuous at } c \iff g(c) = \lim_{x\to c^+} h(x)$$
• Two Different Formulas Meeting at 0: Compute Each Side with Its Own Tool · The two recurring one-sided limits
  $$\lim_{x\to 0^-}\frac{1 - \cos 4x}{x^2} = 8 \qquad \lim_{x\to 0^+}\frac{\sqrt{x}}{\sqrt{16 + \sqrt{x}} - 4} = \lim_{x\to 0^+}\left(\sqrt{16 + \sqrt{x}} + 4\right) = 8$$
• Exponential Junctions and the Given Value at 0 · Exponential-denominator junction
  $$\lim_{x\to 0}\frac{p\sin x + q\tan x}{a^{x} - 1} = \frac{p + q}{\log a} \qquad \log a = k\log 2 \iff a = 2^{k}$$
• Two Junctions, Two Unknowns: Set Up a Linear System · The two-seam system
  $$\text{seam } c_1:\ g(c_1) = h(c_1) \qquad \text{seam } c_2:\ h(c_2) = k(c_2) \qquad \Rightarrow\ \text{two linear equations in } a, b$$
• The Squeeze: x² sin(1/x) Is Continuous for Any Coefficient · Squeeze at 0
  $$\left|x^2\sin\frac{1}{x}\right| \le x^2 \to 0 \ \Rightarrow\ \lim_{x\to 0} x^2\sin\frac{1}{x} = 0$$

Watch out for (5)

• Which piece owns the point?
  → One Junction: Left Limit = Right Limit = Value
• Substituting into the surd piece
  → Two Different Formulas Meeting at 0: Compute Each Side with Its Own Tool
• log a = 4 log 2 means a = 16, not a = 8
  → Exponential Junctions and the Given Value at 0
• Applying continuity at only one seam
  → Two Junctions, Two Unknowns: Set Up a Linear System
• Looking for a constraint that is not there
  → The Squeeze: x² sin(1/x) Is Continuous for Any Coefficient

Formulas (5)

• [x] Is Discontinuous at Every Integer — Counting Them · Jumps of the greatest-integer function
  $$\lim_{x\to n^-}[x] = n - 1 \ne n = [n] \quad (n \in \mathbb{Z}) \qquad [kx] \text{ jumps at } x = \frac{m}{k},\ m \in \mathbb{Z}$$
• Signum-Type Jumps: (x − a)/|x − a| and Products with It · The sign factor
  $$\frac{x - a}{|x - a|} = \begin{cases} 1, & x > a \\ -1, & x < a \end{cases} \qquad \text{jump of size } 2 \text{ at } a$$
• Piecewise with [x] and |x|: Check Every Join and Both Endpoints · Continuity at a closed endpoint
  $$\text{left endpoint } a:\ \lim_{x\to a^+} f(x) = f(a) \qquad \text{right endpoint } b:\ \lim_{x\to b^-} f(x) = f(b)$$
• Composites of [x]: Where Does the Inner Function Cross an Integer? · Jumps of a composite
  $$[g(x)] \text{ jumps where } g(x) \in \mathbb{Z} \text{ and } g \text{ crosses it} \qquad [-u] = -[u] - 1 \ (u \notin \mathbb{Z})$$
• When the Other Factor Vanishes at the Jump · A zero swallows a bounded jump
  $$g \text{ continuous at } n,\ g(n) = 0,\ |h| \le M \text{ near } n \ \Rightarrow\ \lim_{x\to n} g(x)h(x) = 0 = g(n)h(n)$$

Watch out for (5)

• Miscounting the negative side
  → [x] Is Discontinuous at Every Integer — Counting Them
• Cancelling |x − 1| against (x − 1)
  → Signum-Type Jumps: (x − a)/|x − a| and Products with It
• Forgetting the closed right endpoint
  → Piecewise with [x] and |x|: Check Every Join and Both Endpoints
• Answering the 'find k' question when no k exists
  → Composites of [x]: Where Does the Inner Function Cross an Integer?
• The exam key that contradicts the mathematics
  → When the Other Factor Vanishes at the Jump