limits

Limits of a functions:

F(x) approaches the limits L as x approach x so,









Note : f(x)= state whether f(x) is continuous :
F(2)=1


Since f(2) lim f(x)
F(x) is not continuous


Theory: if f and g be two continuous function at x=c then :
F+g, f-g, f/g are continuous function.

Limit laws : if L, M, C and K are real numbers and
Lim f(x)=L and Lim g(x)=M then :
1. sum rule lim (f(x)+g(x))=L+M
2. Difference rule lim (f(x)-g(x))=L-M
3. Product rule lim (f(x).g(x))=L.M
4. Constant multiple lim k.f(x)=k.L
5. Quotient rule lim f(x)/g(x)=L/M, M not equal zero.
6. power rule if r and s are integers then lim(f(x))r/s=Lr/s
The derivative of a function f with respect to x is f defined by :

Provide the limit exist if f exist we say f is differtiable function

Example : find if f(x)=x/x-1


Differentiation rules :
1. dc/dx=0, c=const.
2. dxn/dx=n xn-1

f=1.x.x2.x3.x-2 .x-3.x-4
=0.1.2x.3x2.-2x-3.-3x-4.-4x-5
3. if u and v two differentaible functions of x then u+v and u.v is different.

d/dx (u+v)=du/dx+dv/dx
d/dx(uv)=du/dx.dv/dx
d/dx(u/v)=(du/dx.v-u.dv/dx)/u2