Definitions [8]
"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."
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 left of a, i.e.
\[\lim_{x\to a^{-}}f\left(x\right)\] is the expected value of f at x = a.
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.
A function f(x) is said to be continuous at a point x = a, if the following three conditions are satisfied
- f is defined at every point on an open interval containing a.
- \[\lim_{x\to a}f\left(x\right)\] exists.
- \[\lim_{x\to a}f\left(x\right)=f\left(a\right)\].
A function f(x) is said to be discontinuous at x = a if it is not continuous at x = a, i.e.
- \[\lim_{x\to a}f\left(a\right)\] does not exist.
- The left-hand limit and the right-hand limit are not equal.
- \[\lim_{x\to a}f\left(x\right)\neq f\left(a\right)\].
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.
Theorems and Laws [1]
If f(x) ≤ g(x) ≤ h(x) and \[\lim_{x\to a}\mathrm{f}(x)=l=\lim_{x\to a}\mathrm{h}(x)\]
\[\therefore\lim_{x\to a}g(x)=l\]
