Limit, Continuity and Differentiability – JEE Mains Mathematics

• by Zenith LMS
• Updated April 2025

1. Real-Valued Functions

• A function that maps real numbers to real numbers.
• Examples include polynomial, rational, trigonometric, logarithmic, and exponential functions.

2. Algebra of Functions

• Sum, difference, product, and quotient of functions can be formed:
• If f(x) and g(x) are functions, then (f + g)(x) = f(x) + g(x), (f − g)(x) = f(x) − g(x), (f * g)(x) = f(x) * g(x), and (f/g)(x) = f(x) / g(x), for g(x) ≠ 0.

3. Types of Functions

• Polynomial Functions: Functions involving powers of x, e.g., f(x) = ax² + bx + c
• Rational Functions: Functions of the form f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials
• Trigonometric Functions: f(x) = sin(x), cos(x), tan(x), etc.
• Logarithmic and Exponential Functions: e.g., f(x) = log(x), f(x) = e^x
• Inverse Functions: Functions that reverse the effect of a given function, e.g., if f(x) = y, then f⁻¹(y) = x.

4. Limits

• Limit of a function describes the behavior of the function as the input approaches a particular value.
• Mathematical Representation: lim_{x → a} f(x) = L
• Fundamental limit properties: lim_{x → a} [f(x) ± g(x)] = lim_{x → a} f(x) ± lim_{x → a} g(x)
• Limits are used to define continuity and differentiability.

5. Continuity

• A function is continuous at a point if the following are true:
• f(a) is defined
• lim_{x → a} f(x) exists
• lim_{x → a} f(x) = f(a)
• Continuous functions do not have jumps, breaks, or holes in their graphs.

6. Differentiability

• A function is differentiable at a point if the derivative exists at that point.
• f'(x) = lim_{h → 0} [f(x+h) − f(x)]/h
• If a function is differentiable at a point, it is also continuous at that point.

7. Differentiation of Various Functions

• Sum, Difference, Product, and Quotient: Differentiation rules for sum, difference, product, and quotient of two functions.
• Trigonometric Functions:
  • d/dx [sin(x)] = cos(x)
  • d/dx [cos(x)] = −sin(x)
  • d/dx [tan(x)] = sec²(x)
• Inverse Trigonometric Functions:
  • d/dx [arcsin(x)] = 1/√(1−x²)
  • d/dx [arccos(x)] = −1/√(1−x²)
• Logarithmic Functions:
  • d/dx [ln(x)] = 1/x
  • d/dx [logₐ(x)] = 1/(x ln(a))
• Exponential Functions:
  • d/dx [e^x] = e^x
  • d/dx [a^x] = a^x ln(a)
• Composite Functions:
  • d/dx [f(g(x))] = f'(g(x)) * g'(x)
• Implicit Functions: Differentiation of functions involving both x and y using implicit differentiation.

8. Applications of Derivatives

• Rate of Change: Derivatives are used to find the rate at which one quantity changes with respect to another.
• Monotonic Functions: A function is increasing if its derivative is positive, decreasing if its derivative is negative.
• Maxima and Minima:
  • Use first and second derivative tests to find local maxima and minima of a function.
  • If f'(x) = 0 and f''(x) > 0, the function has a local minimum at x.
  • If f'(x) = 0 and f''(x) < 0, the function has a local maximum at x.