# 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 x_{0}.

**3**. All laws of derivative are applicable on limit.

**4**. If f(x) ≤ g(x) for every x∈X in some deleted neighborhood of x_{0} 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 x_{0}.

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=x_{0}, 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 x_{0} if for each ε> for every δ>0 s.t. x_{0}<x<x_{0}+δ

⇒|f(x)-f(x _{0})|<ε

#### Left continuous:

A function is right continuous if at x_{0} if for each ε> for every δ>0 s.t. x_{0}-δ < x < x_{0} 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:

- 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: