Functions, Limits and continuity

Results

1. If X and y have 'n' and 'm' distinct elements respectively then the number of mappings from X to y = $m^{n}$

Working Rules for finding $D_{f}$ and $R_{f}$

I. For $D_{f}$

1. Solve for 'Y' in terms of x

2. Find those values of c which 'y' is well defined and real. The set of values of x for which 'y' is well defined and real is domain of f ie $D_{f}$

II. For $R_{f}$

For Range we repeat steps 9I) and (ii) with roots of x and y interchanged.

Def. Limit

if the value of f approaches "I' as 'x' approaches 'a' then we say that the limit of f(x) is I and write it as ${}_{X \to a}^{\lim}{f (x) = I .}$

Method to find limit.

1. If f(a) is finite, then ${}_{X \to a}^{\lim}{f (x)}$ =f (a)

2. If f ( x) is a rational function, then remove the common factors between numerator and denominator and then apply the limit.

3. If f(x)) is a irrational function then multiply by the conjugate on numerator and denominator. Then remove the common factors between numerator and denominator if any and apply the limit.

Results

1. ${}_{x \to a}^{\lim}{\frac{x^{n} - a^{n}}{x - a}}$ =n $a^{n - 1}$	2. ${}_{θ \to 0}^{\lim}{\frac{\sin θ}{θ}}$ =1
3. ${}_{θ \to 0}^{\lim}{Sin θ}$ =0	4. ${}_{θ \to 0}^{\lim}{Cos θ}$ =1
5. ${}_{θ \to 0}^{\lim}{\frac{\cos θ}{θ}}$ =0	6. ${}_{θ \to 0}^{\lim}{\frac{\tan θ}{θ}}$ =1
7. ${}_{x \to a}^{\lim}{(1 + x)}^{1 / x}$ =e, ${}_{x \to \infty}^{\lim}{(1 + \frac{1}{x})}^{x}$ =e	8. ${}_{x \to 0}^{\lim}{\frac{e^{x} - 1}{x}}$ =1
9. ${}_{x \to 0}^{\lim}{\frac{a^{x} - 1}{x}}$ = $\log_{e} a$	10. ${}_{x \to 0}^{\lim}{\frac{\log (1 + x)}{x}}$ =1
11. ${}_{θ \to 0}^{\lim}{\frac{\sin^{- 1} x}{x}}$ =1	${}_{θ \to 0}^{\lim}{\frac{\tan^{- 1} x}{x}}$ =1

Indeterminate form

For X= a, if f(x) takes any of the following forms.

$\frac{0}{0}$ , $\frac{\infty}{\infty}$ , $\infty - \infty$ , 0. $\infty$ , $0^{0}$ , $1^{\infty}$ , $\infty^{0}$ Then f(x) is said to be indeterminate.

L-Hospital's Rule

${}_{x \to a}^{\lim}{\frac{f (x)}{g (x)}}$ = ${}_{x \to a}^{\lim}{\frac{F' (x)}{g' (x)}}$ [if f(a)=g(a)=0.

= ${}_{x \to a}^{\lim}{\frac{f " (x)}{g " (x)}}$ [if f'(a)= g'(a)=0]

Left Hand and Right hand Limit

Let f (x) be a function of x. If f(x) $\to 0 as x \to a^{-}$ then 'I' is called the left hand limit of 'f' at 'a' can write as

L. H. L = ${}_{x \to a^{-}}^{\lim}{f (x)}$ =1

Similarly if f(x) $\to$ $a^{+}$ then 'I' is called the right hand limit of 'f" at 'a' and can be write as

R. H. L= ${}_{x \to a^{+}}^{\lim}{f (x)}$ =1

If L. H. L = R. H.L = 1 then we say the limit of the function is 'I'

Continuity at a point

Let f be a real function 'f' is said to be continuous at a real number 'a' if (1) f(a) is defined

(2) ${}_{x \to a}^{\lim}{f (x)}$ exist

(3) ${}_{x \to a}^{\lim}{f (x)}$ =f(a)

Continuous functions

A real function f(x) is said to be a continuous function if it is continuous at every point of its domain.

Discontinuity at a point

A function f is said to be discontinuous at x= a if f is no continuous at x= a ie ${}_{x \to a}^{\lim}{f (x) \neq f (a)}$