Understanding Limits and Boundaries

Limits typically refer to the concept that something can be as close as possible to a given value, but never actually reaches the value. For example, the function 1/x as x approaches infinity.
The concept of a boundary in mathematical terms usually refers to a demarcation line. This can be the edge of a set, such as a number line where numbers greater than or equal to ten belong to the set, or it can be more abstract, such as a line in a physical theory that separates space into different conditions or phases.

Limit Notation

Limit notation is presented as ‘lim’ followed by the function as the variable (‘x’) approaches a certain value.
For example, lim(x→3) of (x^2 - 9)/(x - 3) would be evaluated as 6.

Certain properties hold for limits. Notably, if ‘c’ is a constant and ‘n’ is a positive integer, lim(x→a) c = c and lim(x→a) x = a hold.
Also, limits can be operated with basic arithmetic operations. If ‘m’ and ‘n’ are the limits of two functions, the limits of the sum, difference, product, and quotient of the two functions will be the sum, difference, product, and quotient of ‘m’ and ‘n’, respectively.

An infinite limit means that as the input to the function approaches a certain value, the output of the function grows without bound.
For example, as x approaches 0 from the positive side in the function 1/x, the output goes to infinity.

There are various techniques to evaluate limits including direct substitution, factoring, rationalising, and applying the rules of l’Ôpital in the case of indeterminate forms.

A Boundary Value Problem (BVP) for a given differential equation involves finding a solution satisfying the differential equation and also satisfying two given boundary conditions.
These are common in physics and engineering. For example, a second-order linear differential equation with given boundary conditions can describe the motion of a vibrating string.