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Limits, Continuity, Differentiability

Short Notes

Limits

The limit of a function f(x) as x approaches a point c is the value that f(x) approaches as x gets arbitrarily close to c.

\[\lim_{x \to c} f(x) = L\]
  • Left-hand limit (LHL): \(\lim_{x \to c^-} f(x)\)
  • Right-hand limit (RHL): \(\lim_{x \to c^+} f(x)\)
  • Limit exists iff LHL = RHL

Continuity

A function f(x) is continuous at x = c if:

  1. f(c) is defined
  2. \(\lim_{x \to c} f(x)\) exists
  3. \(\lim_{x \to c} f(x) = f(c)\)

Differentiability

A function is differentiable at x = c if: $\(f'(c) = \lim_{h \to 0} \frac{f(c+h) - f(c)}{h}\)$ exists.

Key Relationship: $\(\text{Differentiable} \Rightarrow \text{Continuous} \Rightarrow \text{Limit exists}\)$ (Converse is NOT always true!)

Key Theories & Formulas

1. Standard Limits

Limit Value
\(\lim_{x \to 0} \frac{\sin x}{x}\) 1
\(\lim_{x \to 0} \frac{\tan x}{x}\) 1
\(\lim_{x \to 0} \frac{1 - \cos x}{x^2}\) \(\frac{1}{2}\)
\(\lim_{x \to 0} \frac{e^x - 1}{x}\) 1
\(\lim_{x \to 0} \frac{\ln(1+x)}{x}\) 1
\(\lim_{x \to 0} \frac{a^x - 1}{x}\) \(\ln a\)
\(\lim_{x \to 0} (1 + x)^{1/x}\) \(e\)
\(\lim_{x \to \infty} (1 + \frac{1}{x})^x\) \(e\)
\(\lim_{x \to 0} \frac{\sin^{-1} x}{x}\) 1
\(\lim_{x \to 0} \frac{\tan^{-1} x}{x}\) 1

2. L'Hôpital's Rule

For indeterminate forms \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\): $\(\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}\)$

Other indeterminate forms: \(0 \cdot \infty\), \(\infty - \infty\), \(0^0\), \(1^\infty\), \(\infty^0\) (convert to \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\))

3. Types of Discontinuity

Type Description
Removable Limit exists but ≠ f(c), or f(c) undefined
Jump LHL ≠ RHL (both finite)
Infinite At least one limit is infinite
Oscillatory Limit doesn't exist (e.g., \(\sin(1/x)\) at x=0)

4. Derivative Formulas

Function Derivative
\(x^n\) \(nx^{n-1}\)
\(e^x\) \(e^x\)
\(a^x\) \(a^x \ln a\)
\(\ln x\) \(\frac{1}{x}\)
\(\sin x\) \(\cos x\)
\(\cos x\) \(-\sin x\)
\(\tan x\) \(\sec^2 x\)

5. Non-Differentiable Cases

A function is NOT differentiable at x = c if:

  • There's a corner/cusp (e.g., |x| at x = 0)
  • There's a vertical tangent
  • There's a discontinuity
  • Left and right derivatives differ

Example Problems

Problem 1: Evaluate \(\lim_{x \to 0} \frac{\sin 5x}{3x}\)

Solution: $\(\lim_{x \to 0} \frac{\sin 5x}{3x} = \lim_{x \to 0} \frac{\sin 5x}{5x} \cdot \frac{5}{3} = 1 \cdot \frac{5}{3} = \mathbf{\frac{5}{3}}\)$

Problem 2: Check continuity of \(f(x) = \begin{cases} \frac{\sin x}{x} & x \neq 0 \\ k & x = 0 \end{cases}\) at x = 0

Solution:

  • \(\lim_{x \to 0} \frac{\sin x}{x} = 1\)
  • For continuity: k = 1

Problem 3: Is \(f(x) = |x|\) differentiable at x = 0?

Solution:

  • Left derivative: \(\lim_{h \to 0^-} \frac{|h| - 0}{h} = \lim_{h \to 0^-} \frac{-h}{h} = -1\)
  • Right derivative: \(\lim_{h \to 0^+} \frac{|h| - 0}{h} = \lim_{h \to 0^+} \frac{h}{h} = 1\)
  • Left ≠ Right → Not differentiable at x = 0

Hardest GATE Questions

Topic: L'Hôpital's Rule with \(1^\infty\) Form

Question (GATE 2018 Style): Evaluate \(\lim_{x \to 0} (\cos x)^{1/x^2}\)

Solution:

  1. Form: \(1^\infty\) (indeterminate)
  2. Let \(y = (\cos x)^{1/x^2}\)
  3. \(\ln y = \frac{\ln(\cos x)}{x^2}\)

  4. Apply L'Hôpital's Rule: $\(\lim_{x \to 0} \frac{\ln(\cos x)}{x^2} = \lim_{x \to 0} \frac{-\tan x}{2x} = -\frac{1}{2}\)$

  5. \(\ln y = -\frac{1}{2}\)

  6. \(y = e^{-1/2} = \mathbf{\frac{1}{\sqrt{e}}}\)

Why it's hard: Requires recognizing \(1^\infty\) form and using logarithmic transformation.

Topic: Differentiability at a Point

Question (GATE 2016 Variant): For what value of n is \(f(x) = x^n \sin(1/x)\) for \(x \neq 0\), and \(f(0) = 0\), differentiable at x = 0?

Solution:

  1. Check derivative at x = 0: $\(f'(0) = \lim_{h \to 0} \frac{h^n \sin(1/h)}{h} = \lim_{h \to 0} h^{n-1} \sin(1/h)\)$

  2. For this limit to exist:

  3. \(|h^{n-1} \sin(1/h)| \leq |h|^{n-1}\)
  4. Need \(n - 1 > 0\), i.e., n > 1
  5. For n = 2: limit = 0 (derivative exists)
  6. For n = 1: limit oscillates (derivative doesn't exist)

Answer: \(\mathbf{n > 1}\) (or n ≥ 2 for integer n)

Topic: Continuity with Floor Function

Question: Find the number of points where \(f(x) = [x] + \sqrt{x - [x]}\) is discontinuous in [0, 3].

Solution:

  1. \(f(x) = [x] + \sqrt{\{x\}}\) where {x} is the fractional part
  2. \(\sqrt{\{x\}}\) is continuous everywhere
  3. [x] is discontinuous at integers: 1, 2, 3
  4. At x = 1: LHL = 0 + 1 = 1, RHL = 1 + 0 = 1 → Continuous
  5. At x = 2: LHL = 1 + 1 = 2, RHL = 2 + 0 = 2 → Continuous
  6. At x = 3: LHL = 2 + 1 = 3, RHL = 3 + 0 = 3 → Continuous
  7. 0 discontinuities in [0, 3]

References