Algebra of Limits Let f and g be two real functions with domain D. We define four new functions f ± g , f/g, fg on domain D by setting (f ± g) (x) = f(x) ± g(x), (fg) (x) = f(x) g(x) (f/g) (x) = f(x)/g(x), if g(x) ≠ 0 for any x ϵ Read more about Algebra of Limits[…]
Method of Evaluating Algebraic Limits Method of evaluating Algebraic Limits when variable tends to Infinity: To evaluate this type of limits e follow the following procedure: Step I: Write down the given expression in form of a rational function, i.e, , if it is not so. Step II: If K is highest power of x Read more about Method of Evaluating Algebraic Limits[…]
Differentiability at a Point Let f(x) be a real valued function defied on an open interval (a, b) and let c ϵ (a, b). Then f(x) is said to be differentiable or derivable at x = c, if exist finitely. This limit is called the derivative or differential coefficient of the function f(x) at x Read more about Differentiability at a Point[…]
Continuity at A Point We say a function f(x) is continuous at a point x = a it means at a point (a, f (a)). The graph of the function has no holes or graphs. That is, its graph is unbroken at point (a, f (a)). The continuity at x = a₂ can be destroyed if Read more about Continuity at A Point[…]
Relations between Roots and Coefficients If α₁, α₂, … αn are roots of the equation f(x) = aₒ xⁿ + a₁ x ⁿˉ¹ + a₂ x ⁿˉ² +. . . + an = 0 then f(x) = aₒ(x – α₁) (x – α₂) ….. (x – αn) aₒ xⁿ + a₁ x ⁿˉ¹ + a₂ Read more about Relations between Roots and Coefficients[…]
Roots of a Quadratic Equation with Real Coefficients An equation of the form ax² + bx + c = 0 … (i) Where a ≠ 0, a, b, c є R is called quadratic equation with real coefficients. The quality D = b² – 4ac is known as the discriminant of the quadratic equation in Read more about Roots of a Quadratic Equation with Real Coefficients[…]
Transformation of Equation Transformation equation 1: Transformation of an equation into another equation whose roots are the reciprocals of the roots of the given equation. Let f(x) = aₒ xⁿ + a₁ x ⁿˉ¹ + a₂ x ⁿˉ² +. . . + an … (1) Be the given equation. Let x and y be respectively Read more about Transformation of Equation[…]
Conjugate Points and Conjugate Lines Conjugate Points: Two points P and Q are said to be conjugate points with respect to the circle S = 0 if Q lies on the polar of P. The condition that the two points P(x₁, y₁) and Q(x₂, y₂) are conjugate points with respect to the circle S = Read more about Conjugate Points and Conjugate Lines[…]
Chord of Contact and Chord Bisected at a given Point: The chord joining the points of contact of the two tangents to a conic drawn from a given point, outside it, is called the chord of contact of tangents.The equation of the chord of contact of tangents drawn from a point (x₁, y₁) to the Read more about Chord of Contact and Chord[…]
General Term in Binomial Expansion The binomial expansion is (x + a) ⁿ = ⁿC₀ xⁿ + ⁿC₁ xⁿ⁻¹ a + ⁿC₂ xⁿ⁻² a² + … + ⁿCr xⁿ⁻r ar + … + ⁿCnx⁰aⁿ. (r + 1)th term is ⁿCr xⁿ⁻r ar. ∴ Tr₊₁ = ⁿCr xⁿ⁻r ar. General term for (x – a )ⁿ Read more about General Term in Binomial Expansion[…]