Kerala State Board New Syllabus Plus One Maths Notes Chapter 13 Limits and Derivatives.

## Kerala Plus One Maths Notes Chapter 13 Limits and Derivatives

Calculus is that branch of mathematics which mainly deals with the study of change in the value of a function as the points in the domain changes.

I. Limit

Limit of a function f(x) at x = a is the behaviors of f(x) at x = a.

x → a^{–}: Means that ‘x’ takes values less than ‘a’ but not ‘a’.

x → a^{+}: Means that ‘x’ takes values greater than ‘a’ but not ‘a’.

x → a: Read as ‘x’ tends to ‘a’, means that ‘x’ takes values very close to ‘a’ but not ‘a’.

\(\lim _{x \rightarrow a^{-}} f(x)=A\): Read as left limit of f(x) is ‘A’, means that f(x) → A as x → a^{–}. To evaluate the left limit we use the following substitution \(\lim _{x \rightarrow a^{-}} f(x)=\lim _{h \rightarrow 0} f(a-h)\)

\(\lim _{x \rightarrow a^{+}} f(x)=B\): Read as right limit of f(x) is ‘B’, means that f(x) → B as x → a^{+}. To evaluate the left limit we use the following substitution \(\lim _{x \rightarrow a^{+}} f(x)=\lim _{h \rightarrow 0} f(a+h)\).

If left limit and right limit of f(x) at x = a are equal, then we say that the limit of the function f(x) exists at x = a and is denoted

by lim \(\lim _{x \rightarrow a} f(x)\). Otherwise we say that \(\lim _{x \rightarrow a} f(x)\) does not exist.

II. Evaluation Methods

- Direct substitution method
- Factorisation method
- Rationalisation method
- Using standard results.

III. Algebra of Limits:

For functions f and g the following holds;

IV. Standard Results

\(\lim _{x \rightarrow a} k=k\), where k is constant.

\(\lim _{x \rightarrow a} f(x)=f(a)\), if f(x) is a polynomial function.

1. \(=\frac{0}{0}\), if possible we can factorise the numerator and denominator and then, cancel the common factors and again put x = a. This factorization method is not possible in all cases so we are studying some standard limits.

V. Derivatives

A derivative of f at a: Suppose f is a real-valued function and a is a point in its domain of definition. The derivative of f at a is defined by \(\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}\)

Provided this limit exists. A derivative of f (x) at a is denoted by f'(a).

Derivative of f at x. Suppose f is a real-valued function, the function defined by \(\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\)

Wherever this limit exists is defined as the derivative of f at x and is denoted by f”(x) |\(\frac{d y}{d x}\)| |y_{1}| y’. This definition of derivative is also called the first principle of the derivative.

VI. Algebra of Derivatives

For functions f and g are differentiable following holds;

VII. Standard Results