JEE Mains & Advanced Preparation Resources

The limit of a function as x approaches a value (a) is the value that the function 'approaches' as x gets closer and closer to a. It's an important concept in calculus for understanding the behavior of functions near certain points.

L'Hopital's Rule is a method for evaluating limits of indeterminate forms (0/0 or ∞/∞). It states that the limit of the ratio of two functions is equal to the limit of the ratio of their derivatives, provided that the latter limit exists. This is a powerful tool to simplify limits.

The vector relation r × a = r × b describes a situation where two vectors, 'a' and 'b', have the same cross product with a vector 'r'. This implies that 'a' and 'b' are either parallel or coincident, since the cross product of parallel vectors is zero.

The shortest distance between two non-parallel lines in 3D space is found by calculating the perpendicular distance from a point on one line to the other line. This method involves finding the projection of a vector connecting the lines onto a vector perpendicular to both lines.

Leibniz Rule is a formula for finding the derivative of a definite integral with respect to its upper or lower limit. It expresses the derivative as the integrand evaluated at the upper limit multiplied by the derivative of the upper limit (or the negative of the integrand evaluated at the lower limit multiplied by the derivative of the lower limit).

A sequence of numbers where each term is found by adding a constant value (called the common difference) to the previous term.

A sequence of numbers where each term is found by multiplying the previous term by a constant value (called the common ratio).

A formula used to calculate a specific term in an arithmetic progression. The formula is: a_n = a + (n-1)d, where a is the first term, d is the common difference, and n is the term number.

A formula used to calculate a specific term in a geometric progression. The formula is: a_n = a * r^(n-1), where a is the first term, r is the common ratio, and n is the term number.

A method used to sum an arithmetic progression. The formula is: S_n = (n/2)*(a + l), where S_n is the sum of the first n terms, a is the first term, and l is the last term.

A method used to sum a geometric progression. The formula is: S_n = a*(1-r^n)/(1-r), where S_n is the sum of the first n terms, a is the first term, and r is the common ratio.

A powerful inequality that compares the arithmetic mean (average) and geometric mean of a set of non-negative numbers. The inequality states that the arithmetic mean is always greater than or equal to the geometric mean.

Methods employed to solve different types of problems involving sequences and series. These might include the Sigma notation method, the Vn method, and the method of differences, among others.