Class 11 Math Chapter 12 Limits and Derivatives Notes (Handwritten & Short Notes )

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Class 11 Math Limits and Derivatives Notes

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Chapter 13: Statistics Notes

Key Points

Limits and Derivatives

Let $y = f(x)$ be a function of $x$x. When at $x = a$ , $f(x)$ takes indeterminate form, then we consider the values of the function which is very near to $a$a. If these values tend to a definite unique number as $x$ tends to $a$, then the unique number so obtained is called the limit of $f(x)$ at $x = a$ and we write it as:

Algebra of limits

Limits of trigonometric functions

Let $f$ and $g$ be two real valued functions with the same domain such that
$f(x) \le g(x)$ for all $x$ in the domain of definition.
For some $a$, if both

Sandwich Theorem

Let $f$, $g$ and $h$ be real functions such that
$f(x) \le g(x) \le h(x)$ for all $x$ in the common domain of definition.

For some real number $a$, if

Derivatives

Suppose $f$ is a real valued function and $a$

is a point in its domain of definition. The derivative of $f$f at $a$a is defined by:$$\lim_{h \to 0} \frac{f(a + h) – f(a)}{h}$$

provided this limit exists. Derivative of $f(x)$f at $a$a is denoted by $f'(a)$.

Suppose $f$ is a real valued function. The function defined by:$$\lim_{h \to 0} \frac{f(x + h) – f(x)}{h}$$

wherever the limit exists, is defined to be the derivative of $f$ at $x$ and is denoted by $f'(x)$. This definition of derivative is also called the first principle of derivative.