## Lecture-2

Some of the limits used in this lecture:

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

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

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

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

Play Video

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

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

3. $$f(x)=\left\{\begin{array}{l} {kx+1, {\rm if}\, \, x\le 5} \\ {3x-5, {\rm if}\, \, x>5} \end{array}\right.$$

4. $$f(x)=\left\{\begin{array}{l} {\frac{x^{2} -2x-3}{x+1} , x\ne -1} \\ {k, x=-1} \end{array}\right.$$

5. $$f(x)=\left\{\begin{array}{l} {\frac{k\cos x}{\pi -2x} , {\rm if}\, \, x\ne \frac{\pi }{2} } \\ {3, {\rm if}\, \, x=\frac{\pi }{2} } \end{array}\right.$$

6. $$f(x)=\left\{\begin{array}{l} {5, {\rm if}\, \, x\le 2} \\ {ax+b, {\rm if}\, \, 2

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

8. $$f(x)=\left\{\begin{array}{l} {\frac{\log (1+ax)-\log (1-bx)}{x} , x\ne 0} \\ {k, x=0} \end{array}\right.$$

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

10. $$f(x)=\left\{\begin{array}{l} {\frac{1-\cos 4x}{x^{2} } , x<0} \\ {k, x=0} \\ {\frac{\sqrt{x} }{\sqrt{16+\sqrt{x} } -4} , x>0} \end{array}\right.$$

11. $$f(x)=\left\{\begin{array}{l} {\frac{x}{|x|+2x^{2} } , x\ne 0} \\ {k, x=0} \end{array}\right.$$

बिना संघर्ष कोई महान नही होता,
बिना कुछ किये जय जय कार नही होता,
जब तक नहीं पड़ती हथोड़े की चोट,
तब तक कोई पत्थर भी लोगों के लिए भगवान नही होता।