Important theorems in limit,continuity & discontinuity

Hey there!Hope you have read about the limit & continuity. In this context we discuss about the important & useful theorems of limit which are useful in learning, also, we discuss about the continuity,discontinuity with its pros & cons…

1. Uniqueness theorem of lim:

If limx→x0 f(x) exist, then it is unique.i.e. The limit of every function is unique.

2. If limx→x0 =l,then f is bounded on some deleted neighborhood of x0.

3. All laws of derivative are applicable on limit.

4. If f(x) ≤ g(x) for every x∈X in some deleted neighborhood of x0 then
limx→x0 f(x) ≤ limx→x0 g(x)

5. If g(x) ≤ f(x) ≤ h(x), for every x∈X in some deleted neighborhood of x0.
If limx→x0 g(x)=limx→x0 h(x)=l
then,limx→x0 f(x)=l.

Some important trigonometric limit theorems:
6.limθ→θ0 sinθ/θ=1

7.limθ→θ0 tanθθ=1

8. limθ→θ0 (1+x)1/x=e, Where, e is an irrational no. its value lies between 2&3.

9. limθ→θ0 (ax-1)/x = log a

10. limθ→θ0 (ax-1)/x = 1

11. limθ→θ =log(1+x)/x = 1

12. log a+log b = log (a.b)

13. log a – log b = log (a/b)

14. log ab =b.log a

15. logab = log a/log b =1/logab

16. log a =1

17.log a =0

18. log 0 =log (1/e) = not defined.

Continuity:

A function is continuous at a point if its right & left limits are defined and are equal at that point.
A function is continuous at x=x 0, if lim x→x 0= f(x 0).

ε-δ definition of continuity:

A function is continuous at a point x=x0, if for any ε>0 , there exist  a δ>o s.t.
|x – x 0|<δ ⇒ |f(x) – f(x 0)|<ε

Right continuous:

A function is right continuous if at x0 if for each ε> for every δ>0 s.t. x0<x<x0
⇒|f(x)-f(x 0)|<ε

Left continuous:

A function is right continuous if at x0 if for each ε> for every δ>0 s.t. x0-δ < x < x0 f(x 0)=f(x 0) i.e. for every ε>0 ∃ δ>0
⇒|f(x)-f(x 0)|<ε

Continuity on an interval:

A function is continuous on an interval I, if it is continuous on every points of I.

Some tips about continuity of functions:

  • The functions sin x, cos x, e,a are continuous at every real number.
  • The polynomial function is continuous at every real number.
  • log x is continuous at every x>0.
  • If two functions f(x) & g(x) are continuous at x=a, then  [ f(x)+g(x) ; f(x)-g(x);  f(x)*g(x); f(x)/g(x) ] all are continuous.

Discontinuity:

If a function f is not continuous at some point x, then f is discontinuous at x.
There are different cases of discontinuity:

  1. f(x).f(x),f(x+)all exist but are not all equal.
  2. f(x+) or f(x-) do not exist.
  3. f(x+) ≠ f(x-)
  4. f(x+) = f(x-) ≠ f(x)

Cases 1,2,4 give rise to discontinuity of type 1 or simple discontinuity.
Case 2 give rise to discontinuity of type 2
Case 3 gives rise to ordinary discontinuity
Case 4 gives removable discontinuity

Various types of discontinuity and their sequences are explained in the diagram below:

 

discontinuity & types