Some of the limits used in this lecture:

$\theta {\rm in\; "radians"\; not\; in\; degrees}$
$$\lim_{\theta \to 0} \frac{\sin \theta }{\theta } =1$$

$$\lim_{\theta \to 0} \frac{\tan \theta }{\theta } =1$$
$$\lim_{\theta \to 0} \frac{1-\cos \theta }{\theta } =0$$

$$\lim_{x \to 0} \frac{e^{x} -1}{x} =1$$
$$\lim_{x \to 0} \frac{\log _{e} \, (1+x)}{x} =1$$

$$\lim_{x \to a} \frac{x^{n} -a^{n} }{x-a} =na^{n-1}$$
$$\lim_{\theta \to 0} \frac{a^{x} -1}{x} =\log _{e} a$$

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### Part - 1

In this lecture, I am discussing about how to check continuity and discontinuity in a graph of a function through limits. I am using my own assignments to explain NCERT Exercise 5.1 Questions as well as Extra (HOTS) question for explanation and CBSE Board exam point of view. You can download assignments and all other material related to maths on my website.
Questions Discussed in this lecture:

1. $f(x)=\left\{\begin{array}{l} {2x+3,\, \, \, {\rm if}\, \, x\le 2} \\ {2x-3,\, \, \, {\rm if}\, \, x>2} \end{array}\right.$

2. $f(x)=\left\{\begin{array}{l} {\frac{|x|}{x} , {\rm if}\, \, x\ne 0} \\ {0, {\rm if}\, \, x=0} \end{array}\right.$

3. $f(x)=\left\{\begin{array}{l} {x+1, {\rm if}\, \, x\ge 1} \\ {x^{2} +1, {\rm if}\, \, x<1} \end{array}\right.$

4. $f(x)=\left\{\begin{array}{l} {x^{10} -1, \, \, {\rm if}\, \, x\le 1} \\ {x^{2} , \, \, {\rm if}\, \, x>1} \end{array}\right.$

5. $f(x)=\left\{\begin{array}{l} {\frac{x}{|x|} , {\rm if}\, \, x<0} \\ {-1, {\rm if}\, \, x\ge 0} \end{array}\right.$

6. $f(x)=\left\{\begin{array}{l} {x^{3} -3, {\rm if}\, \, x\le 2} \\ {x^{2} +1, {\rm if}\, \, x>2} \end{array}\right.$

7. $f(x)=\left\{\begin{array}{l} {x+5, {\rm if}\, \, x\le 1} \\ {x-5, {\rm if}\, \, x>1} \end{array}\right.$

8. $f(x)=\left\{\begin{array}{l} {|x|+3, \, \, {\rm if}\, \, x\le -3} \\ {-2x, \, \, {\rm if}\, \, -3

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### Part - 2

9. $f(x)=\left\{\begin{array}{l} {3,\, \, {\rm if}\, \, 0\le x\le 1} \\ {4,\, \, {\rm if}\, \, 1

10. $f(x)=\left\{\begin{array}{l} {2x, {\rm if}\, \, x<0} \\ {0, {\rm if}\, \, 0\le x\le 1} \\ {4x, {\rm if}\, \, x>1} \end{array}\right.$

11. $f(x)=\left\{\begin{array}{l} {-2, {\rm if}\, \, x\le -1} \\ {2x, {\rm if}\, \, -11} \end{array}\right.$

12. $f(x)=\left\{\begin{array}{l} {\frac{e^{x} -1}{\log (1+2x)} , x\ne 0} \\ {7, x=0} \end{array}\right.$

13. $f(x)=\left\{\begin{array}{l} {\frac{\sin x}{x} , {\rm if}\, \, x<0} \\ {x+1, {\rm if}\, \, x\ge 0} \end{array}\right.$

14. $f(x)=\left\{\begin{array}{l} {x^{2} \sin \frac{1}{x} , {\rm if}\, \, x\ne 0} \\ {0, {\rm if}\, \, x=0} \end{array}\right.$

15. $f(x)=\left\{\begin{array}{l} {\sin x-\cos x, {\rm if}\, \, x\ne 0} \\ {-1, {\rm if}\, \, x=0} \end{array}\right.$

16. $f(x)=\left\{\begin{array}{l} {\frac{\sin 3x}{\tan 2x} , x<0} \\ {\frac{3}{2} , x=0} \\ {\frac{\log (1+3x)}{e^{2x} -1} , x>0} \end{array}\right.$

जल को बर्फ़ में बदलने में वक्त लगता है,
सूरज को निकलने में वक्त लगता है,
किस्मत को तो हम बदल नही सकते,
लेकिन अपने हौसलो से किस्मत बदलने में वक्त लगता है।