Revision: Differential Calculus - Limits and Continuity Mathematics HSC Science Class 11 Tamil Nadu Board of Secondary Education

"dy/dx is called the derivative of y with respect to x (which is the rate of change of y with respect to change in x) and the process of finding the derivative is called differentiation."

When we have the values of f near x to the left of a, i.e.
\[\lim_{x\to a^{-}}f\left(x\right)\] is the expected value of f at x = a.

If f(x) approaches a real number l, when x approaches a, then l is called the limit of f(x).

Symbolically, \[\lim_{x\to a}f\left(x\right)=l\]

When we have the values of f near x to the right of a i.e.
\[\lim_{x\to a^{+}}f\left(x\right)\] is the expected value of f at x = a.

If \[\lim_{x\to a^{-}}f\left(x\right)=\lim_{x\to a^{+}}f\left(x\right)\neq f\left(a\right),\] then f(x) is said to be removable discontinuous.

If \[\lim_{x\to a^{+}}f\left(x\right)\neq\lim_{x\to a^{-}}f\left(x\right),\] then f(x) is said to be non-removable discontinuous.

A function f(x) is said to be continuous at a point x = a, if the following three conditions are satisfied

1. f is defined at every point on an open interval containing a.
2. \[\lim_{x\to a}f\left(x\right)\] exists.
3. \[\lim_{x\to a}f\left(x\right)=f\left(a\right)\].

The principle which states that for a non-viscous liquid in streamline flow passing through a tube of varying cross-section, the product of the area of cross-section and the velocity of flow remains constant at every point is called the Equation of Continuity.

For a non-viscous liquid in streamline flow passing through a tube of varying cross-section:

av = constant

or equivalently:

a ∝ \[\frac {1}{v}\]

where: 

• a = area of cross-section of the tube
• v = velocity of flow of the liquid

If f(x) ≤ g(x) ≤ h(x) and \[\lim_{x\to a}\mathrm{f}(x)=l=\lim_{x\to a}\mathrm{h}(x)\]

\[\therefore\lim_{x\to a}g(x)=l\]