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
Some important trigonometric limit theorems:
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 a =1
17.log 1 a =0
18. log 0 =log (1/e) = not defined.
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
A function is right continuous if at x0 if for each ε> for every δ>0 s.t. x0<x<x0+δ
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
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.
If a function f is not continuous at some point x, then f is discontinuous at x.
There are different cases of discontinuity:
- f(x).f(x–),f(x+)all exist but are not all equal.
- f(x+) or f(x-) do not exist.
- f(x+) ≠ f(x-)
- 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: