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Calculus

Limit & Continuity

Mathematics Core
20 min read
IAT Core Framework
High Yield

1. Core Concept

A Limit describes the value a function approaches as the input gets infinitely close to a specific point (from both left and right). Continuity means the function's actual value at that point perfectly matches this approaching limit—visually, you can draw the graph without lifting your pen.

2. Key Formulas & Standard Limits

Existence & Continuity

  • Existence of Limit: LHL = RHL.
    limx→a‾ f(x) = limx→a+ f(x) = L
  • Condition for Continuity: Limit exists AND equals the function's value.
    limx→a f(x) = f(a)

Trigonometric Limits (x in radians)

limx→0
sin xx
= 1
limx→0
tan xx
= 1
limx→0
1 - cos xx2
=
12

Exponential & Logarithmic Limits

limx→0
ex - 1x
= 1
limx→0
ax - 1x
= ln(a)
limx→0
ln(1 + x)x
= 1

Algebraic Limit

limx→a
xn - anx - a
= n · an-1
limx→a f(x)g(x) = elimx→a [f(x) - 1] · g(x)
The 1∞ Form Shortcut: Applies only if lim f(x) = 1 and lim g(x) = ∞.

3. Important Results

  • Algebra of Limits & Continuity: If f(x) and g(x) are continuous at x = a, then f ± g, f · g, and f / g (provided g(a) ≠ 0) are also continuous at x = a.
  • Composition: If g is continuous at a and f is continuous at g(a), then f(g(x)) is continuous at a.
  • Universally Continuous Functions: Polynomials, exponentials (ex), sine, cosine, and absolute value |x| are continuous everywhere (ℝ).
  • Domain-Specific Continuity: Rational functions, logarithmic functions, and inverse trigonometric functions are continuous strictly within their domains.

Types of Discontinuity

  • Removable: Limit exists but f(a) is different or not defined.
  • Jump: LHL ≠ RHL.
  • Infinite: Function goes to ±∞.
  • Oscillatory: Function keeps oscillating (e.g., sin(1/x)).

Intermediate Value Theorem (IVT)

If a function is continuous on [a, b] and takes values f(a) and f(b), then it takes every value between them.

Implication: If f(a) and f(b) have opposite signs, there exists at least one root in (a, b).

4. Conceptual Insights

  • LHL vs RHL Necessity: You only need to check Left-Hand Limit and Right-Hand Limit separately when the function changes definition at x = a (e.g., piecewise functions, modulus |x-a|, Greatest Integer Function [x], or fractional part {x}).
  • Sandwich (Squeeze) Theorem: If g(x) ≤ f(x) ≤ h(x) near a, and limx→a g(x) = limx→a h(x) = L, then limx→a f(x) = L. Highly useful for limits involving oscillating terms like sin(1/x).
  • L'Hôpital's Rule: If lim[f(x)/g(x)] yields a 0/0 or ∞/∞ form, you can differentiate the numerator and denominator separately: lim [f'(x) / g'(x)].

5. Common Mistakes

  • Blind L'Hôpital: Applying L'Hôpital's Rule when the limit is NOT in a 0/0 or ∞/∞ indeterminate form. Always plug in the limit value first to check!
  • Ignoring the Domain: Trying to find limx→0 √x using both LHL and RHL. The LHL doesn't exist over real numbers (since √(negative) is imaginary). The limit is evaluated only in the domain (x ≥ 0).
  • GIF Continuity: The Greatest Integer Function f(x) = [x] is discontinuous at every integer. Students often forget to check integer points when [x] is part of a larger equation.

6. Example Applications

Finding Unknowns in Piecewise Functions

Scenario: A function is given as ax + b for x < 1, 3 at x = 1, and cx2 for x > 1. You are told it is continuous everywhere.

Solution: Immediately set LHL = Value = RHL at the breakpoint.
a(1) + b = 3 = c(1)2

Limit of x·sin(1/x) as x → 0

Insight: As x → 0, sin(1/x) oscillates wildly between -1 and 1. However, it is bounded. Multiplying a bounded function by something going to 0 (which is x) results in 0.

Proven formally by Squeeze Theorem: -|x| ≤ x·sin(1/x) ≤ |x|.

7. IAT Exam Focus Points

  • The 1∞ Form: This is the most frequently tested specific limit type in JEE/IAT. Memorize the elim (f(x)-1)·g(x) shortcut formula completely.
  • Continuity at Breakpoints: Be prepared to equate LHL and RHL for functions involving |x|, [x], or e1/x to find the value of a constant k that makes the function continuous.
  • L'Hôpital with Chain Rule: Exam questions often combine limits with integration using Leibniz's Rule. If you see an integral in the numerator of a limit, differentiate it using the Fundamental Theorem of Calculus while applying L'Hôpital's.
  • Standard Limit Manipulation: Often, you need to multiply and divide by specific terms to "create" the standard limits (e.g., turning sin(5x) / x into 5 ·[sin(5x) / 5x] = 5 · 1 = 5).

8. Practice Mock Test

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Take a quick 15-question assessment specifically designed for Limit & Continuity. Challenge yourself with IAT-level questions.

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Limit & Continuity

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