गणित को गणित की एक कुंजी के रूप में प्रयोग करके प्रकृति के चलन की व्याख्या में सफलतापूर्वक लागू किया जा सकता है - व्हाइटहेड

12.1 परिचय

यह अध्याय गणित के गणित का परिचय है। गणित उस शाखा है जो मुख्य रूप से एक फलन के मान में परिवर्तन का अध्ययन करती है जब डोमेन में बिंदुओं में परिवर्तन होता है। पहले, हम व्युत्पन्न के एक भावनात्मक विचार देते हैं (इसे वास्तव में परिभाषित किए बिना)। फिर हम सीमा की एक सादगी परिभाषा देते हैं और कुछ सीमाओं के बैलेंस का अध्ययन करते हैं। फिर हम व्युत्पन्न की परिभाषा पर वापस आते हैं और व्युत्पन्नों के कुछ बैलेंस का अध्ययन करते हैं। हम कुछ मानक फलनों के व्युत्पन्न भी प्राप्त करते हैं।

सर आइजैक न्यूटन (1642-1727 ईसा पूर्व)

12.2 व्युत्पन्नों का भावनात्मक विचार

भौतिक प्रयोगों ने पुष्टि की है कि एक ऊँची खोए से गिरते हुए शरीर को $(1642-1727)$ मीटर की दूरी $4.9 t^{2}$ सेकंड में $t$ मीटर की दूरी $s$ मीटर की दूरी $t$ सेकंड में $s=4.9 t^{2}$ के रूप में दी गई है।

आसन्न तालिका 13.1 ऊँची खोए से गिरते हुए शरीर की विभिन्न समय अंतरालों में मीटर में दूरी देती है।

इस डेटा से $t=2$ सेकंड में शरीर की वेग $t=2$ सेकंड में $t=2$ सेकंड की वेग $t=t_1$ और $t=t_2$ के बीच $t=t_l$ और $t=t_2$ सेकंड $(t_2-t_1)$ के बीच $t=1$ और $t=2$ के बीच $t=t_1$ और $t=2$ के बीच $t_1$ के लिए $(v), t=t_1$ सेकंड $t=2$ सेकंड के बीच $t$ $s$ के बीच $t_1$ $v$ के बीच $t=2$ के बीच $t=2$ के बीच $t=2$ सेकंड $19.551 m / s$ के बीच $t=2$ सेकंड से शुरू होने वाले विभिन्न समय अंतरालों के लिए $v$ $t=2$ $t=t_2$ के बीच $v$ मीटर प्रति सेकंड $t=2$ $t_2$ के बीच $t_2$ $v$ के बीच $t=2$ $t=2$ के बीच $t=2$ $t=2$ के बीच $t=2$ $t=2$ के बीच $t=2$ $19.551 m / s$ $19.649 m / s$ के बीच $t=2$ $19.551 m / s$ $19.649 m / s$ के बीच $s=4.9 t^{2}$ $t=2$ $s$ $t$ $h_1, h_2, \ldots$ $C_1 B_1=s_1-s_0$ $h_1=AC_1$ $A$ $v(t)$ $t=2$ $s=4.9 t^{2}$ $t=2$ $f(x)=x^{2}$ $x$ $f(x)$ $f(x)$ $x$ $f(x)$ $x$ $f(x)$ $x=0$ $x \to a, f(x) \to l$ $l$ $f(x)$ $\lim\limits_{x \to a} f(x)=l$ $g(x)=|x|, x \neq 0$ $g(0)$ $g(x)$ $x$ $g(x)$ $\lim\limits_{x \to 0} g(x)=0$ $y=|x|$ $x \neq 0$ $h(x)$ $x$ $y=h(x)$ $x=a$ $x$ $a$ $x$ $a$ $x$ $a$ $a$ $a$ $f(x)$ $f(x)$ $f(x)$ $x$ $a$ $f$ $f(x)$ $x \leq 0$ $f(x)$ $f$ $f(x)$ $x>0$ $f(x)$ $f(x)$ $x$ $\lim\limits_{x \to a^{-}} f(x)$ $f$ $x=a$ $f$ $x$ $a$ $f$ $a$ $\lim\limits_{x \to a^{+}} f(x)$ $f$ $x=a$ $f$ $x$ $a$ $f(x)$ $a$ $f(x)$ $x=a$ $\lim\limits_{x \to a} f(x)$ $f(x)=x+10$ $x=5$ $f(x)$ $x$ $4.9,4.95,4.99,4.995 \ldots$ $x$ $f(x)$ $f(x)$ $x=5$ $x=4.995$ $f(x)$ $x=5$ $x$ $f(x)$ $f(x)$ $f(x)$ $x$ $f(x)=x+10$ $(5,15)$ $x=5$ $f(x)=x^{3}$ $x=1$ $f(x)$ $x$ $x$ $f(x)$ $f(x)$ $x=1$ $x=0.999$ $f(x)$ $x=1$ $x$ $f(x)$ $f(x)$ $f(x)$ $x$ $f(x)=x^{3}$ $(1,1)$ $x=1$ $f(x)=3 x$ $x=2$ $x$ $f(x)$ $x$ $f(x)$ $x=2$ $x=2$ $f(x)=3$ $x=2$ $f(x)=3$ $x$ $(0,3)$ $\lim\limits_{x \to a} f(x)=3$ $a$ $f(x)=x^{2}+x$ $\lim\limits_{x \to 1} f(x)$ $f(x)$ $x=1$ $x$ $f(x)$ $\lim\limits_{x \to 1^{-}} f(x)=\lim\limits_{x \to 1^{+}} f(x)=\lim\limits_{x \to 1} f(x)=2$ $f(x)=x^{2}+x$ $x$ $(1,2)$ $f(x)=\sin x$ $\lim\limits_{x \to \frac{\pi}{2}} \sin x$ $f(x)$ $\frac{\pi}{2}$ $f(x)=\sin x$ $\lim\limits_{x \to \frac{\pi}{2}} \sin x=1$ $x$ $\frac{\pi}{2}-0.1$ $\frac{\pi}{2}-0.01$ $\frac{\pi}{2}+0.01$ $\frac{\pi}{2}+0.1$ $f(x)$ $f(x)=x+\cos x$ $\lim\limits_{x \to 0} f(x)$ $f(x)$ $x$ $f(x)$ $\lim\limits_{x \to 0} f(x)=f(0)=1$ $f(x)=\frac{1}{x^{2}}$ $x>0$ $\lim\limits_{x \to 0} f(x)$ $f(x)$ $x$ $x$ $n$ $x$ $0, f(x)$ $f(x)$ $x$ $10^{-n}$ $f(x)$ $10^{2 n}$ $\lim\limits_{x \to 0} f(x)$ $x$ $f(x)$ $x$ $x-2$ $x+2$ $x$ $f(x)$ $x=0$ $x=0$ $\lim\limits_{x \to 1} f(x)$ $x$ $f(x)$ $f(x)$ $x$ $f(x)$ $x$ $x=1$ $f(x)$ $f(x)$ $x$ $f$ $g$ $\lim\limits_{x \to a} f(x)$ $\lim\limits_{x \to a} g(x)$ $g$ $g(x)=\lambda$ $\lambda$ $f$ $n f(x)=a_0+a_1 x+a_2 x^{2}+\ldots+a_n x^{n}$ $a_1$ $a_n \neq 0$ $n$ $\lim\limits_{x \to a} x=a$ $n$ $f(x)=a_0+a_1 x+a_2 x^{2}+\ldots+a_n x^{n}$ $a_0, a_1 x, a_2 x^{2}, \ldots, a_n x^{n}$ $f$ $f(x)=\frac{g(x)}{h(x)}$ $g(x)$ $h(x)$ $h(x) \neq 0$ $h(a)=0$ $g(a) \neq 0$ $g(a)=0$ $g(x)=(x-a)^{k} g_1(x)$ $k$ $(x-a)$ $g(x)$ $h(x)=(x-a)^{l} h_1(x)$ $h(a)=0$ $k>l$ $k<l$ $\lim\limits_{x \to 1}[x^{3}-x^{2}+1] \quad$ $\lim\limits_{x \to 3}[x(x+1)]$ $\lim\limits_{x \to-1}[1+x+x^{2}+\ldots+x^{10}]$ $\lim\limits_{x \to 1}[x^{3}-x^{2}+1]=1^{3}-1^{2}+1=1$ $\lim\limits_{x \to 3}[x(x+1)]=3(3+1)=3(4)=12$ $\lim\limits_{x \to-1}[1+x+x^{2}+\ldots+x^{10}]=1+(-1)+(-1)^{2}+\ldots+(-1)^{10}$ $\lim\limits_{x \to 1}[\frac{x^{2}+1}{x+100}]$ $\lim\limits_{x \to 2}[\frac{x^{3}-4 x^{2}+4 x}{x^{2}-4}]$ $\lim\limits_{x \to 2}[\frac{x^{2}-4}{x^{3}-4 x^{2}+4 x}]$ $\lim\limits_{x \to 2}[\frac{x^{3}-2 x^{2}}{x^{2}-5 x+6}]$ $\lim\limits_{x \to 1}[\frac{x-2}{x^{2}-x}-\frac{1}{x^{3}-3 x^{2}+2 x}]$ $\frac{0}{0}$ $\frac{0}{0}$ $\lim\limits_{x \to 1} \frac{x^{2}+1}{x+100}=\frac{1^{2}+1}{1+100}=\frac{2}{101}$ $\frac{0}{0}$ $\lim\limits_{x \to 2} \frac{x^{3}-4 x^{2}+4 x}{x^{2}-4}=\lim\limits_{x \to 2} \frac{x(x-2)^{2}}{(x+2)(x-2)}$ $\frac{0}{0}$ $\lim\limits_{x \to 2} \frac{x^{2}-4}{x^{3}-4 x^{2}+4 x}=\lim\limits_{x \to 2} \frac{(x+2)(x-2)}{x(x-2)^{2}}$ $\frac{0}{0}$ $\quad \lim\limits_{x \to 2} \frac{x^{3}-2 x^{2}}{x^{2}-5 x+6}=\lim\limits_{x \to 2} \frac{x^{2}(x-2)}{(x-2)(x-3)}$ $\frac{0}{0}$ $\quad \lim\limits_{x \to 1}[\frac{x^{2}-2}{x^{2}-x}-\frac{1}{x^{3}-3 x^{2}+2 x}]=\lim\limits_{x \to 1} \frac{x^{2}-4 x+3}{x(x-1)(x-2)}$ $(x-1)$ $x \neq 1$ $n$ $n$ $a$ $(x^{n}-a^{n})$ $(x-a)$ $\lim\limits_{x \to 1} \frac{x^{15}-1}{x^{10}-1}$ $\lim\limits_{x \to 0} \frac{\sqrt{1+x}-1}{x}$ $y=1+x$ $y \to 1$ $x \to 0$ $f$ $g$ $f(x) \leq g(x)$ $x$ $a$ $\lim\limits_{x \to a} f(x)$ $\lim\limits_{x \to a} g(x)$ $\lim\limits_{x \to a} f(x) \leq \lim\limits_{x \to a} g(x)$ $f$ $g$ $h$ $f(x) \leq g(x) \leq h(x)$ $x$ $a$ $\lim\limits_{x \to a} f(x)=l=\lim\limits_{x \to a} h(x)$ $\lim\limits_{x \to a} g(x)=l$ $\sin (-x)=-\sin x$ $\cos (-x)=\cos x$ $0<x<\frac{\pi}{2}$ $O$ $x$ $0<x<\frac{\pi}{2}$ $CD$ $OA$ $AC$ $\triangle OAC<$ $OAC<$ $\triangle OAB$ $\quad \frac{1}{2} OA . CD<\frac{x}{2 \pi} . \pi .(OA)^{2}<\frac{1}{2} OA . AB$ $\quad CD<x . OA<AB$ $\triangle OCD$ $\quad AB=OA \cdot \tan x$ $0<x<\frac{\pi}{2}, \sin x$ $\sin x$ $1<\frac{x}{\sin x}<\frac{1}{\cos x} . \quad$ $\lim\limits_{x \to 0} \frac{\sin x}{x}=1$ $\lim\limits_{x \to 0} \frac{1-\cos x}{x}=0$ $\frac{\sin x}{x}$ $\cos x$ $\lim\limits_{x \to 0} \cos x=1$ $1-\cos x=2 \sin ^{2}(\frac{x}{2})$ $x \to 0$ $\frac{x}{2} \to 0$ $y=\frac{x}{2}$ $\lim\limits_{x \to 0} \frac{\sin 4 x}{\sin 2 x}$ $\lim\limits_{x \to 0} \frac{\tan x}{x}$ $\lim\limits_{x \to 0} \frac{\tan x}{x}=\lim\limits_{x \to 0} \frac{\sin x}{x \cos x}=\lim\limits_{x \to 0} \frac{\sin x}{x} \cdot \lim\limits_{x \to 0} \frac{1}{\cos x}=1.1=1$ $\lim\limits_{x \to a} \frac{f(x)}{g(x)}$ $f(a)$ $g(a)$ $f(x)=f_1(x) f_2(x)$ $f_1(a)=0$ $f_2(a) \neq 0$ $g(x)=g_1(x) g_2(x)$ $g_1(a)=0$ $g_2(a) \neq 0$ $f(x)$ $g(x)$ $\quad \quad \quad \quad\quad \quad \quad \frac{f(x)}{g(x)}=\frac{p(x)}{q(x)}, \text{ where } q(x) \neq 0 $ $\quad \quad \quad \quad \quad \lim\limits_{x \to a} \frac{f(x)}{g(x)}=\frac{p(a)}{q(a)}$ $f$ $f$ $a$ $f(x)$ $f^{\prime}(a)$ $f^{\prime}(a)$ $f(x)$ $a$ $x$ $x=2$ $f(x)=3 x$ $3 x$ $x=2$ $f(x)=2 x^{2}+3 x-5$ $x=-1$ $f^{\prime}(0)+3 f^{\prime}(-1)=0$ $f(x)$ $x=-1$ $x=0$ $\quad \quad \quad \quad f^{\prime}(0)+3 f^{\prime}(-1)=0 $ $\sin x$ $x=0$ $f(x)=\sin x$ $f(x)=3$ $x=0$ $x=3$ $\quad f^{\prime}(3)=\lim\limits_{h \to 0} \frac{f(3+h)-f(3)}{h}=\lim\limits_{h \to 0} \frac{3-3}{h}=0$ $y=f(x)$ $f(\boldsymbol{{}a}+\boldsymbol{{}h})$ $P=(a, f(a))$ $Q=(a+h, f(a+h)$ $f^{\prime}(a)=\lim\limits_{h \to 0} \frac{f(a+h)-f(a)}{h}$ $PQR$ $\tan (QPR)$ $PQ$ $h$ $Q$ $P$ $PQ$ $P$ $y=f(x)$ $f$ $f$ $f$ $f$ $x$ $f^{\prime}(x)$ $f^{\prime}(x)$ $f^{\prime}(x)$ $\frac{d}{d x}(f(x))$ $y=f(x)$ $\frac{d y}{d x}$ $f(x)$ $y$ $x$ $D(f(x))$ $f$ $x=a$ $.\frac{d}{dx} f(x)| _ {a} $ $.\frac{df}{dx}| _ {a} $ $(\frac{d f}{d x}) _ {x=a}$ $f(x)=10 x$ $f^{\prime}(x)=\lim\limits_{h \to 0} \frac{f(x+h)-f(x)}{h}$ $f(x)=x^{2}$ $f^{\prime}(x)=\lim\limits_{h \to 0} \frac{f(x+h)-f(x)}{h}$ $f(x)=a$ $a$ $f^{\prime}(x)=\lim\limits_{h \to 0} \frac{f(x+h)-f(x)}{h}$ $f(x)=\frac{1}{x}$ $f^{\prime}(x)=\lim\limits_{h \to 0} \frac{f(x+h)-f(x)}{h}$ $f$ $g$ $u=f(x)$ $v=g(x)$ $f(x)=x$ $f^{\prime}(x)=\lim\limits_{h \to 0} \frac{f(x+h)-f(x)}{h}=\lim\limits_{h \to 0} \frac{x+h-x}{h}$ $f(x)=10 x=x+\ldots .+x$ $(i)$ $f(x)=10 x=u v$ $u$ $v(x)=x$ $f(x)=10 x=u v$ $u$ $v(x)=x$ $f(x)=x^{2}$ $f(x)=x^{2}=x . x$ $f(x)=x^{n}$ $n x^{n-1}$ $n$ $(x+h)^{n}=({ }^{n} C_0) x^{n}+({ }^{n} C_1) x^{n-1} h+\ldots+({ }^{n} C_n) h^{n}$ $(x+h)^{n}-x^{n}=h(n x^{n-1}+\ldots+h^{n-1})$ $n$ $n=1$ $x$ $n$ $f(x)=a_n x^{n}+a {n-1} x^{n-1}+\ldots .+a_1 x+a_0$ $a_i s$ $a_n \neq 0$ $6 x^{100}-x^{55}+x$ $600 x^{99}-55 x^{54}+1$ $f(x)=1+x+x^{2}+x^{3}+\ldots+x^{50}$ $x=1$ $1+2 x+3 x^{2}+\ldots+50 x^{49}$ $x=1$ $1+2(1)+3(1)^{2}+\ldots+50(1)^{49}=1+2+3+\ldots+50=\frac{(50)(51)}{2}=1275$ $f(x)=\frac{x+1}{x}$ $x=0$ $u=x+1$ $v=x$ $u^{\prime}=1$ $v^{\prime}=1$ $\sin x$ $f(x)=\sin x$ $\tan x$ $f(x)=\tan x$ $f(x)=\sin ^{2} x$ $f$ $f$ $f(x)=\frac{2 x+3}{x-2}$ $f(x)=x+\frac{1}{x}$ $x=2$ $f^{\prime}$ $x=2$ $x=0$ $f^{\prime}$ $x=0$ $f(x)$ $f(x)$ $\sin x+\cos x$ $x \sin x$ $f^{\prime}(x)=\frac{f(x+h)-f(x)}{h}$ $\quad f^{\prime}(x)=\lim\limits{h \to 0} \frac{f(x+h)-f(x)}{h}=\lim\limits_{h \to 0} \frac{(x+h) \sin (x+h)-x \sin x}{h}$ $f(x)=\sin 2 x$ $g(x)=\cot x$ $\sin 2 x=2 \sin x \cos x$ $g(x)=\cot x=\frac{\cos x}{\sin x}$ $\frac{d g}{d x}=\frac{d}{d x}(\cot x)=\frac{d}{d x}(\frac{\cos x}{\sin x})$ $\cot x=\frac{1}{\tan x}$ $\tan x$ $\sec ^{2} x$ $\frac{x^{5}-\cos x}{\sin x}$ $\frac{x+\cos x}{\tan x}$ $h(x)=\frac{x^{5}-\cos x}{\sin x}$ $\frac{x+\cos x}{\tan x}$ $f$ $a, \lim\limits_{x \to a} f(x)$ $f(a)$ $f$ $g$ $f$ $a$ $f$ $x$ $u$ $v$ $\frac{\sin \alpha}{\alpha}$ $\alpha=0$ $\frac{\Delta y}{\Delta x}=\frac{f(x+i)-f(x)}{i}$ $i \to 0$ $y^{\prime}$ $f^{\prime}(x)$ https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/images/ncertbook/math/m11/limits_and_derivatives/ncert_m11_ch12_issac.png" https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/images/ncertbook/math/m11/limits_and_derivatives/ncert_m11_ch12_fig_12_1.png" https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/images/ncertbook/math/m11/limits_and_derivatives/ncert_m11_ch12_fig_12_2.png" https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/images/ncertbook/math/m11/limits_and_derivatives/ncert_m11_ch12_fig_12_3.png" https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/images/ncertbook/math/m11/limits_and_derivatives/ncert_m11_ch12_fig_12_4.png" https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/images/ncertbook/math/m11/limits_and_derivatives/ncert_m11_ch12_fig_12_5.png" https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/images/ncertbook/math/m11/limits_and_derivatives/ncert_m11_ch12_fig_12_6.png" https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/images/ncertbook/math/m11/limits_and_derivatives/ncert_m11_ch12_fig_12_7.png" https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/images/ncertbook/math/m11/limits_and_derivatives/ncert_m11_ch12_fig_12_8.png" https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/images/ncertbook/math/m11/limits_and_derivatives/ncert_m11_ch12_fig_12_9.png" https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/images/ncertbook/math/m11/limits_and_derivatives/ncert_m11_ch12_fig_12_10.png" https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/images/ncertbook/math/m11/limits_and_derivatives/ncert_m11_ch12_fig_12_11.png" $$ \begin{aligned} & =\frac{\text{ Distance travelled between } t_2=2 \text{ and } t_1=0}{\text{ Time interval }(t_2-t_1)} \\ & =\frac{(19.6-0) m}{(2-0) s}=9.8 m / s . \end{aligned} $$ $$ \frac{(19.6-4.9) m}{(2-1) s}=14.7 m / s $$ $$ \lim\limits_{x \to 5^{-}} f(x)=15 . $$ $$ \lim\limits_{x \to 5^{+}} f(x)=15 \text{. } $$ $$ \lim\limits_{x \to 5^{-}} f(x)=\lim\limits_{x \to 5^{+}} f(x)=\lim\limits_{x \to 5} f(x)=15 . $$ $$ \lim\limits_{x \to 1^{-}} f(x)=1 \text{. } $$ $$ \lim\limits_{x \to 1^{+}} f(x)=1 \text{. } $$ $$ \lim\limits_{x \to 1^{-}} f(x)=\lim\limits_{x \to 1^{+}} f(x)=\lim\limits_{x \to 1} f(x)=1 . $$ $$ \lim\limits_{x \to 2^{-}} f(x)=\lim\limits_{x \to 2^{+}} f(x)=\lim\limits_{x \to 2} f(x)=6 $$ $$ \lim\limits_{x \to 0^{-}} f(x)=-2 $$ $$ \lim\limits_{x \to 0^{+}} f(x)=2 $$ $$ f(x)=\begin{cases} x+2 & x \neq 1 \\ 0 & x=1 \end{cases} $$ $$ \cos x<\frac{\sin x}{x}<1 \quad \text{ for } 0<|x|<\frac{\pi}{2} \quad \quad \quad \quad \quad ({}^*) $$ $$ f^{\prime}(0)=\lim\limits_{h \to 0} \frac{f(0+h)-f(0)}{h}=\lim\limits_{h \to 0} \frac{3-3}{h}=\lim\limits_{h \to 0} \frac{0}{h}=0 . $$ $$ \frac{d}{d x}[f(x)+g(x)]=\frac{d}{d x} f(x)+\frac{d}{d x} g(x) . $$ $$ \frac{d}{d x}[f(x)-g(x)]=\frac{d}{d x} f(x)-\frac{d}{d x} g(x) $$ $$ \frac{d}{d x}[f(x) \cdot g(x)]=\frac{d}{d x} f(x) \cdot g(x)+f(x) \cdot \frac{d}{d x} g(x) $$ $$ \frac{d}{d x}(\frac{f(x)}{g(x)})=\frac{\frac{d}{d x} f(x) \cdot g(x)-f(x) \frac{d}{d x} g(x)}{(g(x))^{2}} $$ $$ \begin{aligned} \frac{d f}{d x} & =\frac{d}{d x}(x \cdot x)=\frac{d}{d x}(x) \cdot x+x \cdot \frac{d}{d x}(x) \\ & =1 \cdot x+x \cdot 1=2 x . \end{aligned} $$ $(1642-1727)$ $4.9 t^{2}$ $t$ $s$ $t$ $s=4.9 t^{2}$ $t=2$ $t=2$ $t=2$ $t=t_1$ $t=t_2$ $t=t_l$ $t=t_2$ $(t_2-t_1)$ $t=1$ $t=2$ $t=t_1$ $t=2$ $t_1$ $(v), t=t_1$ $t=2$ $t$ $s$ $t_1$ $v$ $t=2$ $t=2$ $t=2$ $19.551 m / s$ $t=2$ $v$ $t=2$ $t=t_2$ $v$ $t=2$ $t_2$ $t_2$ $v$ $t=2$ $t=2$ $t=2$ $t=2$ $t=2$ $t=2$ $t=2$ $19.551 m / s$ $19.649 m / s$ $t=2$ $19.551 m / s$ $19.649 m / s$ $s=4.9 t^{2}$ $t=2$ $s$ $t$ $h_1, h_2, \ldots$ $C_1 B_1=s_1-s_0$ $h_1=AC_1$ $A$ $v(t)$ $t=2$ $s=4.9 t^{2}$ $t=2$ $f(x)=x^{2}$ $x$ $f(x)$ $f(x)$ $x$ $f(x)$ $x$ $f(x)$ $x=0$ $x \to a, f(x) \to l$ $l$ $f(x)$ $\lim\limits_{x \to a} f(x)=l$ $g(x)=|x|, x \neq 0$ $g(0)$ $g(x)$ $x$ $g(x)$ $\lim\limits_{x \to 0} g(x)=0$ $y=|x|$ $x \neq 0$ $h(x)$ $x$ $y=h(x)$ $x=a$ $x$ $a$ $x$ $a$ $x$ $a$ $a$ $a$ $f(x)$ $f(x)$ $f(x)$ $x$ $a$ $f$ $f(x)$ $x \leq 0$ $f(x)$ $f$ $f(x)$ $x>0$ $f(x)$ $f(x)$ $x$ $\lim\limits_{x \to a^{-}} f(x)$ $f$ $x=a$ $f$ $x$ $a$ $f$ $a$ $\lim\limits_{x \to a^{+}} f(x)$ $f$ $x=a$ $f$ $x$ $a$ $f(x)$ $a$ $f(x)$ $x=a$ $\lim\limits_{x \to a} f(x)$ $f(x)=x+10$ $x=5$ $f(x)$ $x$ $4.9,4.95,4.99,4.995 \ldots$ $x$ $f(x)$ $f(x)$ $x=5$ $x=4.995$ $f(x)$ $x=5$ $x$ $f(x)$ $f(x)$ $f(x)$ $x$ $f(x)=x+10$ $(5,15)$ $x=5$ $f(x)=x^{3}$ $x=1$ $f(x)$ $x$ $x$ $f(x)$ $f(x)$ $x=1$ $x=0.999$ $f(x)$ $x=1$ $x$ $f(x)$ $f(x)$ $f(x)$ $x$ $f(x)=x^{3}$ $(1,1)$ $x=1$ $f(x)=3 x$ $x=2$ $x$ $f(x)$ $x$ $f(x)$ $x=2$ $x=2$ $f(x)=3$ $x=2$ $f(x)=3$ $x$ $(0,3)$ $\lim\limits_{x \to a} f(x)=3$ $a$ $f(x)=x^{2}+x$ $\lim\limits_{x \to 1} f(x)$ $f(x)$ $x=1$ $x$ $f(x)$ $\lim\limits_{x \to 1^{-}} f(x)=\lim\limits_{x \to 1^{+}} f(x)=\lim\limits_{x \to 1} f(x)=2$ $f(x)=x^{2}+x$ $x$ $(1,2)$ $f(x)=\sin x$ $\lim\limits_{x \to \frac{\pi}{2}} \sin x$ $f(x)$ $\frac{\pi}{2}$ $f(x)=\sin x$ $\lim\limits_{x \to \frac{\pi}{2}} \sin x=1$ $x$ $\frac{\pi}{2}-0.1$ $\frac{\pi}{2}-0.01$ $\frac{\pi}{2}+0.01$ $\frac{\pi}{2}+0.1$ $f(x)$ $f(x)=x+\cos x$ $\lim\limits_{x \to 0} f(x)$ $f(x)$ $x$ $f(x)$ $\lim\limits_{x \to 0} f(x)=f(0)=1$ $f(x)=\frac{1}{x^{2}}$ $x>0$ $\lim\limits_{x \to 0} f(x)$ $f(x)$ $x$ $x$ $n$ $x$ $0, f(x)$ $f(x)$ $x$ $10^{-n}$ $f(x)$ $10^{2 n}$ $\lim\limits_{x \to 0} f(x)$ $x$ $f(x)$ $x$ $x-2$ $x+2$ $x$ $f(x)$ $x=0$ $x=0$ $\lim\limits_{x \to 1} f(x)$ $x$ $f(x)$ $f(x)$ $x$ $f(x)$ $x$ $x=1$ $f(x)$ $f(x)$ $x$ $f$ $g$ $\lim\limits_{x \to a} f(x)$ $\lim\limits_{x \to a} g(x)$ $g$ $g(x)=\lambda$ $\lambda$ $f$ $n f(x)=a_0+a_1 x+a_2 x^{2}+\ldots+a_n x^{n}$ $a_1$ $a_n \neq 0$ $n$ $\lim\limits_{x \to a} x=a$ $n$ $f(x)=a_0+a_1 x+a_2 x^{2}+\ldots+a_n x^{n}$ $a_0, a_1 x, a_2 x^{2}, \ldots, a_n x^{n}$ $f$ $f(x)=\frac{g(x)}{h(x)}$ $g(x)$ $h(x)$ $h(x) \neq 0$ $h(a)=0$ $g(a) \neq 0$ $g(a)=0$ $g(x)=(x-a)^{k} g_1(x)$ $k$ $(x-a)$ $g(x)$ $h(x)=(x-a)^{l} h_1(x)$ $h(a)=0$ $k>l$ $k<l$ $\lim\limits_{x \to 1}[x^{3}-x^{2}+1] \quad$ $\lim\limits_{x \to 3}[x(x+1)]$ $\lim\limits_{x \to-1}[1+x+x^{2}+\ldots+x^{10}]$ $\lim\limits_{x \to 1}[x^{3}-x^{2}+1]=1^{3}-1^{2}+1=1$ $\lim\limits_{x \to 3}[x(x+1)]=3(3+1)=3(4)=12$ $\lim\limits_{x \to-1}[1+x+x^{2}+\ldots+x^{10}]=1+(-1)+(-1)^{2}+\ldots+(-1)^{10}$ $\lim\limits_{x \to 1}[\frac{x^{2}+1}{x+100}]$ $\lim\limits_{x \to 2}[\frac{x^{3}-4 x^{2}+4 x}{x^{2}-4}]$ $\lim\limits_{x \to 2}[\frac{x^{2}-4}{x^{3}-4 x^{2}+4 x}]$ $\lim\limits_{x \to 2}[\frac{x^{3}-2 x^{2}}{x^{2}-5 x+6}]$ $\lim\limits_{x \to 1}[\frac{x-2}{x^{2}-x}-\frac{1}{x^{3}-3 x^{2}+2 x}]$ $\frac{0}{0}$ $\frac{0}{0}$ $\lim\limits_{x \to 1} \frac{x^{2}+1}{x+100}=\frac{1^{2}+1}{1+100}=\frac{2}{101}$ $\frac{0}{0}$ $\lim\limits_{x \to 2} \frac{x^{3}-4 x^{2}+4 x}{x^{2}-4}=\lim\limits_{x \to 2} \frac{x(x-2)^{2}}{(x+2)(x-2)}$ $\frac{0}{0}$ $\lim\limits_{x \to 2} \frac{x^{2}-4}{x^{3}-4 x^{2}+4 x}=\lim\limits_{x \to 2} \frac{(x+2)(x-2)}{x(x-2)^{2}}$ $\frac{0}{0}$ $\quad \lim\limits_{x \to 2} \frac{x^{3}-2 x^{2}}{x^{2}-5 x+6}=\lim\limits_{x \to 2} \frac{x^{2}(x-2)}{(x-2)(x-3)}$ $\frac{0}{0}$ $\quad \lim\limits_{x \to 1}[\frac{x^{2}-2}{x^{2}-x}-\frac{1}{x^{3}-3 x^{2}+2 x}]=\lim\limits_{x \to 1} \frac{x^{2}-4 x+3}{x(x-1)(x-2)}$ $(x-1)$ $x \neq 1$ $n$ $n$ $a$ $(x^{n}-a^{n})$ $(x-a)$ $\lim\limits_{x \to 1} \frac{x^{15}-1}{x^{10}-1}$ $\lim\limits_{x \to 0} \frac{\sqrt{1+x}-1}{x}$ $y=1+x$ $y \to 1$ $x \to 0$ $f$ $g$ $f(x) \leq g(x)$ $x$ $a$ $\lim\limits_{x \to a} f(x)$ $\lim\limits_{x \to a} g(x)$ $\lim\limits_{x \to a} f(x) \leq \lim\limits_{x \to a} g(x)$ $f$ $g$ $h$ $f(x) \leq g(x) \leq h(x)$ $x$ $a$ $\lim\limits_{x \to a} f(x)=l=\lim\limits_{x \to a} h(x)$ $\lim\limits_{x \to a} g(x)=l$ $\sin (-x)=-\sin x$ $\cos (-x)=\cos x$ $0<x<\frac{\pi}{2}$ $O$ $x$ $0<x<\frac{\pi}{2}$ $CD$ $OA$ $AC$ $\triangle OAC<$ $OAC<$ $\triangle OAB$ $\quad \frac{1}{2} OA . CD<\frac{x}{2 \pi} . \pi .(OA)^{2}<\frac{1}{2} OA . AB$ $\quad CD<x . OA<AB$ $\triangle OCD$ $\quad AB=OA \cdot \tan x$ $0<x<\frac{\pi}{2}, \sin x$ $\sin x$ $1<\frac{x}{\sin x}<\frac{1}{\cos x} . \quad$ $\lim\limits_{x \to 0} \frac{\sin x}{x}=1$ $\lim\limits_{x \to 0} \frac{1-\cos x}{x}=0$ $\frac{\sin x}{x}$ $\cos x$ $\lim\limits_{x \to 0} \cos x=1$ $1-\cos x=2 \sin ^{2}(\frac{x}{2})$ $x \to 0$ $\frac{x}{2} \to 0$ $y=\frac{x}{2}$ $\lim\limits_{x \to 0} \frac{\sin 4 x}{\sin 2 x}$ $\lim\limits_{x \to 0} \frac{\tan x}{x}$ $\lim\limits_{x \to 0} \frac{\tan x}{x}=\lim\limits_{x \to 0} \frac{\sin x}{x \cos x}=\lim\limits_{x \to 0} \frac{\sin x}{x} \cdot \lim\limits_{x \to 0} \frac{1}{\cos x}=1.1=1$ $\lim\limits_{x \to a} \frac{f(x)}{g(x)}$ $f(a)$ $g(a)$ $f(x)=f_1(x) f_2(x)$ $f_1(a)=0$ $f_2(a) \neq 0$ $g(x)=g_1(x) g_2(x)$ $g_1(a)=0$ $g_2(a) \neq 0$ $f(x)$ $g(x)$ $\quad \quad \quad \quad\quad \quad \quad \frac{f(x)}{g(x)}=\frac{p(x)}{q(x)}, \text{ where } q(x) \neq 0 $ $\quad \quad \quad \quad \quad \lim\limits_{x \to a} \frac{f(x)}{g(x)}=\frac{p(a)}{q(a)}$ $f$ $f$ $a$ $f(x)$ $f^{\prime}(a)$ $f^{\prime}(a)$ $f(x)$ $a$ $x$ $x=2$ $f(x)=3 x$ $3 x$ $x=2$ $f(x)=2 x^{2}+3 x-5$ $x=-1$ $f^{\prime}(0)+3 f^{\prime}(-1)=0$ $f(x)$ $x=-1$ $x=0$ $\quad \quad \quad \quad f^{\prime}(0)+3 f^{\prime}(-1)=0 $ $\sin x$ $x=0$ $f(x)=\sin x$ $f(x)=3$ $x=0$ $x=3$ $\quad f^{\prime}(3)=\lim\limits_{h \to 0} \frac{f(3+h)-f(3)}{h}=\lim\limits_{h \to 0} \frac{3-3}{h}=0$ $y=f(x)$ $f(\boldsymbol{{}a}+\boldsymbol{{}h})$ $P=(a, f(a))$ $Q=(a+h, f(a+h)$ $f^{\prime}(a)=\lim\limits_{h \to 0} \frac{f(a+h)-f(a)}{h}$ $PQR$ $\tan (QPR)$ $PQ$ $h$ $Q$ $P$ $PQ$ $P$ $y=f(x)$ $f$ $f$ $f$ $f$ $x$ $f^{\prime}(x)$ $f^{\prime}(x)$ $f^{\prime}(x)$ $\frac{d}{d x}(f(x))$ $y=f(x)$ $\frac{d y}{d x}$ $f(x)$ $y$ $x$ $D(f(x))$ $f$ $x=a$ $.\frac{d}{dx} f(x)| _ {a} $ $.\frac{df}{dx}| _ {a} $ $(\frac{d f}{d x}) _ {x=a}$ $f(x)=10 x$ $f^{\prime}(x)=\lim\limits_{h \to 0} \frac{f(x+h)-f(x)}{h}$ $f(x)=x^{2}$ $f^{\prime}(x)=\lim\limits_{h \to 0} \frac{f(x+h)-f(x)}{h}$ $f(x)=a$ $a$ $f^{\prime}(x)=\lim\limits_{h \to 0} \frac{f(x+h)-f(x)}{h}$ $f(x)=\frac{1}{x}$ $f^{\prime}(x)=\lim\limits_{h \to 0} \frac{f(x+h)-f(x)}{h}$ $f$ $g$ $u=f(x)$ $v=g(x)$ $f(x)=x$ $f^{\prime}(x)=\lim\limits_{h \to 0} \frac{f(x+h)-f(x)}{h}=\lim\limits_{h \to 0} \frac{x+h-x}{h}$ $f(x)=10 x=x+\ldots .+x$ $(i)$ $f(x)=10 x=u v$ $u$ $v(x)=x$ $f(x)=10 x=u v$ $u$ $v(x)=x$ $f(x)=x^{2}$ $f(x)=x^{2}=x . x$ $f(x)=x^{n}$ $n x^{n-1}$ $n$ $(x+h)^{n}=({ }^{n} C_0) x^{n}+({ }^{n} C_1) x^{n-1} h+\ldots+({ }^{n} C_n) h^{n}$ $(x+h)^{n}-x^{n}=h(n x^{n-1}+\ldots+h^{n-1})$ $n$ $n=1$ $x$ $n$ $f(x)=a_n x^{n}+a {n-1} x^{n-1}+\ldots .+a_1 x+a_0$ $a_i s$ $a_n \neq 0$ $6 x^{100}-x^{55}+x$ $600 x^{99}-55 x^{54}+1$ $f(x)=1+x+x^{2}+x^{3}+\ldots+x^{50}$ $x=1$ $1+2 x+3 x^{2}+\ldots+50 x^{49}$ $x=1$ $1+2(1)+3(1)^{2}+\ldots+50(1)^{49}=1+2+3+\ldots+50=\frac{(50)(51)}{2}=1275$ $f(x)=\frac{x+1}{x}$ $x=0$ $u=x+1$ $v=x$ $u^{\prime}=1$ $v^{\prime}=1$ $\sin x$ $f(x)=\sin x$ $\tan x$ $f(x)=\tan x$ $f(x)=\sin ^{2} x$ $f$ $f$ $f(x)=\frac{2 x+3}{x-2}$ $f(x)=x+\frac{1}{x}$ $x=2$ $f^{\prime}$ $x=2$ $x=0$ $f^{\prime}$ $x=0$ $f(x)$ $f(x)$ $\sin x+\cos x$ $x \sin x$ $f^{\prime}(x)=\frac{f(x+h)-f(x)}{h}$ $\quad f^{\prime}(x)=\lim\limits{h \to 0} \frac{f(x+h)-f(x)}{h}=\lim\limits_{h \to 0} \frac{(x+h) \sin (x+h)-x \sin x}{h}$ $f(x)=\sin 2 x$ $g(x)=\cot x$ $\sin 2 x=2 \sin x \cos x$ $g(x)=\cot x=\frac{\cos x}{\sin x}$ $\frac{d g}{d x}=\frac{d}{d x}(\cot x)=\frac{d}{d x}(\frac{\cos x}{\sin x})$ $\cot x=\frac{1}{\tan x}$ $\tan x$ $\sec ^{2} x$ $\frac{x^{5}-\cos x}{\sin x}$ $\frac{x+\cos x}{\tan x}$ $h(x)=\frac{x^{5}-\cos x}{\sin x}$ $\frac{x+\cos x}{\tan x}$ $f$ $a, \lim\limits_{x \to a} f(x)$ $f(a)$ $f$ $g$ $f$ $a$ $f$ $x$ $u$ $v$ $\frac{\sin \alpha}{\alpha}$ $\alpha=0$ $\frac{\Delta y}{\Delta x}=\frac{f(x+i)-f(x)}{i}$ $i \to 0$ $y^{\prime}$ $f^{\prime}(x)$ https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/images/ncertbook/math/m11/limits_and_derivatives/ncert_m11_ch12_issac.png" https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/images/ncertbook/math/m11/limits_and_derivatives/ncert_m11_ch12_fig_12_1.png" https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/images/ncertbook/math/m11/limits_and_derivatives/ncert_m11_ch12_fig_12_2.png" https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/images/ncertbook/math/m11/limits_and_derivatives/ncert_m11_ch12_fig_12_3.png" https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/images/ncertbook/math/m11/limits_and_derivatives/ncert_m11_ch12_fig_12_4.png" https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/images/ncertbook/math/m11/limits_and_derivatives/ncert_m11_ch12_fig_12_5.png" https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/images/ncertbook/math/m11/limits_and_derivatives/ncert_m11_ch12_fig_12_6.png" https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/images/ncertbook/math/m11/limits_and_derivatives/ncert_m11_ch12_fig_12_7.png" https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/images/ncertbook/math/m11/limits_and_derivatives/ncert_m11_ch12_fig_12_8.png" https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/images/ncertbook/math/m11/limits_and_derivatives/ncert_m11_ch12_fig_12_9.png" https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/images/ncertbook/math/m11/limits_and_derivatives/ncert_m11_ch12_fig_12_10.png" https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/images/ncertbook/math/m11/limits_and_derivatives/ncert_m11_ch12_fig_12_11.png" अध्याय 12 सीमाएँ और व्युत्पन्न