লিমিটের সূত্র (Limit formula) 

লিমিটের সূত্র (Limit formula) 

যেন তেন প্রকারেন ( যেকোনো উপায় ) অসং জ্ঞায়িত অবস্থায় দূর করতে হবে।

প্রয়োজনীয় কিছু ধারা (Some necessary sections) 

1. (1+x)^{n} =1+n x+\frac{n(n-1)}{2 !} \mathrm{x}^{2}+\frac{\mathrm{n}(\mathrm{n}-1)(n-2)}{3 !} x^{3}+-------
2. e^{x} =1+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\frac{x^{4}}{4 !}+
3. \ln (1+x) =x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+------
4. \ln (1-x) =-x-\frac{x^{2}}{2}-\frac{x^{3}}{3}-\frac{x^{4}}{4}-------
5. e^{-x} =1-x+\frac{x^{2}}{2 !}-\frac{x^{3}}{3 !}+\frac{x^{4}}{4 !}-\text { - - - - - - - }
6. a^{x} =1+x \ln a+\frac{x^{2}}{2 !}(\ln a)^{2}+\frac{x^{3}}{3 !}(\ln a)^{3}+----------
7. a^{-x} =1-x \ln a+\frac{x^{2}}{2 !}(\ln a)^{2}-\frac{x^{3}}{3 !}(\ln a)^{3}+-------

বিশেষ কিছু ফাংশনের সীমা নির্ণয় (Determining the limits of certain functions) :

\text { 1. } \frac{\operatorname{tin}_{i i}}{x \rightarrow 0} \frac{e^{x}-1}{x}=1

প্রমানঃ           e^{x}=1+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+-------

\therefore \frac{\lim }{x \rightarrow 0} \frac{e^{x}-1}{x} =\frac{\lim }{x \rightarrow 0} \frac{1}{x}\left\{\left(1+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+----\right)-1\right\}

=\frac{\lim }{x \rightarrow 0} \frac{1}{x}\left\{x+\frac{x^{2}}{2 !}+\frac{x^{3}}{2 !}+---\right\}  

=\frac{\lim }{x \rightarrow 0} \frac{1}{x} \cdot x\left\{1+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+---\right\}

=1  [Proved]

\text { 2. } \frac{\lim }{x \rightarrow \mathbb{0}} \frac{\sin x}{x}=1

প্রমানঃ 

এখানে-

APB চাপ

OP ব্যাসার্ধ

A ও B বিন্দুতে অঙ্কিত  স্পর্শক OP এর বর্ধিতাংশকে c বিন্দুতে পরস্পরকে ছেদ করে।

<P0A=x রেডিয়ান

এখন-

জ্যা AB < চাপ APB  < AC + BC

⟹2AD < 2 চাপ AP < 2AC

⟹AD < চাপ AP <AC

⟹\frac{A D}{O A} <\frac{\text {চাপ } A P}{O A} <\frac{A C}{O A}

 

⟹\sin x <x <tan x--------1

 

⟹1<\frac{x}{\sin x} <\frac{1}{\cos x}

 

⟹\cos x <\frac{\sin x}{x} <1

 

x \rightarrow 0 হলে,\cos x \rightarrow 1

 

⟹\frac{\lim }{x \rightarrow 0} \cos x <\frac{\lim }{x \rightarrow 0} \frac{\sin x}{x} <1

 

⟹1<\frac{\lim }{x \rightarrow 0} \frac{\sin x}{x} <1

 

∴\frac{\lim }{x \rightarrow 0} \frac{\sin x}{x} =1

 

[Proved]

3.\frac{\ln _{\text {iii }}}{x \rightarrow 0} \frac{\operatorname{tar} x}{x}=1

প্রমানঃ 

(1) হতে পাই –

\sin x < x < \tan x

 

⟹\cos x < \frac{x}{\tan x} < 1

 

⟹\frac{\lim }{x \rightarrow 0} \cos x <\frac{\lim }{x \rightarrow 0} \frac{x}{\tan x} <1

 

⟹1<\frac{\lim }{x \rightarrow 0} \frac{x}{\tan x} <1

 

∴\frac{\lim }{x \rightarrow 0} \frac{x}{\tan x} =1

 

[Proved]

 

4.\frac{\lim }{x \rightarrow 0} \frac{\ln (1+x)}{x} =1

 

প্রমানঃ 

\frac{\lim }{x \rightarrow 0} \frac{\ln (1+x)}{x} =\frac{\lim }{x \rightarrow 0} \frac{1}{x} \left(x-\frac{x^{2}}{2}+\frac{x^{3}}{3}+\frac{x^{4}}{4}+----\right)

 

=\frac{\lim }{x \rightarrow 0} \frac{1}{x} .x \left(1-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+-----\right)

 

=1

 

[Proved]

 

5.\frac{\lim }{x \rightarrow 0} \frac{{x^{n}-a^{n}}}{x-a} =n a^{n}-1

প্রমানঃ 

ধরি, x=a+h ; h \rightarrow 0 যখন x \rightarrow a

\frac{\lim }{x \rightarrow a} \frac{x^{n}-a^{n}}{x-a}=\frac{\lim }{h \rightarrow 0} \frac{(a+h)^{\mathrm{n}}-a^{\mathrm{n}}}{a+h-a}

 

\begin{array}{l} =\frac{\lim }{h \rightarrow 0} \frac{(\mathrm{a}+\mathrm{h})^{\mathrm{n}}-\mathrm{a}^{\mathrm{n}}}{\mathrm{h}} \\ =\frac{\lim }{h \rightarrow 0} \frac{1}{h}\left\{a^{n}\left(1+\frac{h}{a}\right)^{n}-a^{n}\right\} \\ =\frac{\lim }{h \rightarrow 0} \frac{a^{n}}{h}\left\{\left(1+\frac{h}{a}\right)^{n}-1\right\} \\ =\frac{\lim }{h \rightarrow 0} \frac{a^{n}}{h}\left\{1+\frac{n h}{a}+\frac{n(n-1)}{2 !} \frac{h^{2}}{a^{2}}+\frac{n(n-1)(n-2)}{3 !} \frac{h^{3}}{a^{3}}+-----1\right\} \\ =\frac{\lim }{h \rightarrow 0} \frac{a^{n}}{h} \cdot \frac{n h}{a}\left\{1+\frac{(n-1)}{2 !} \frac{h}{a}+\frac{n(n-1)(n-2)}{3 !} \frac{h^{2}}{a^{2}}+---\right\} \\ =\frac{n a^{n}}{a}=n a^{n-1} \end{array}

[Proved]

⟹ একনজরে সব সূত্রঃ

1. \frac{\lim }{x \rightarrow 0} \frac{\sin x}{x}=1
2. \frac{\lim }{x \rightarrow 0} \frac{x}{\sin x}=1
3. \frac{\lim }{x \rightarrow 0} \frac{\tan x}{x}=1
4. \frac{\lim }{x \rightarrow 0} \frac{x}{\tan x}=1
5. \frac{\lim }{x \rightarrow 0} \frac{e^{x}-1}{x}=1
6. \frac{\lim }{x \rightarrow 0} \frac{\ln (1+x)}{x}=1
7. \frac{\lim }{x \rightarrow 0} \frac{(1+x)^{n}-1}{x}=n
8. \frac{\lim }{x \rightarrow 0} \frac{x^{n}-a^{n}}{x-a}=n a^{n-1}
9. \frac{\lim }{x \rightarrow 0}(1+x)^{\frac{1}{x}}=e
10. \frac{\lim }{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{x}=e