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

2. If lim_x→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
lim_x→x0 f(x) ≤ lim_x→x0 g(x)

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

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

7.lim_θ→θ0 tanθθ=1

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

9. lim_θ→θ0 (a^x-1)/x = log a

10. lim_θ→θ0 (a^x-1)/x = 1

11. lim_θ→θ =log(1+x)/x = 1

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

13. log a – log b = log (a/b)

14. log a^b =b.log a

15. log^a_b = log a/log b =1/log^a_b

16. log ^a _a =1

17.log ¹ _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+δ
⇒|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: