Limits and Derivatives Lecture 2
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Part - 1

NCERT EXERCISE 13.1

Question 1. \lim_{x \to 3}  x+3

Question 2. \lim_{x \to \pi} \left ( x – \frac{22}{7} \right )

Question 3. \lim_{r \to 1} \pi r^2

Question 4.  \lim_{x \to 4} \frac{4x+3}{x-2}

Question 5. \lim_{x \to -1} \frac{x^{10}+x^5+1}{x-1}

Question 6.  \lim_{x \to 0} \frac{(x+1)^5-1}{x}

Question 7.  \lim_{x \to 2} \frac{3x^2-x-10}{x^2-4}

Question 8.  \lim_{x \to 3} \frac{x^4-81}{2x^2-5x-3}

Question 9.  \lim_{x \to 0} \frac{ax+b}{cx+1}

Question 11.  \lim_{x \to 1} \frac{ax^2+bx+c}{cx^2+bx+a} a+b+c \ne 0

Question 10. \lim_{z \to 1} \frac{z^{\frac{1}{3}}-1}{z^{\frac{1}{6}}-1}

Question 12. \lim_{x \to -2} \frac{\frac{1}{x}+\frac{1}{2}}{x+2}

Example 1.
(i).
\lim_{x \to 1} [x^3-x^2+1]
(ii). \lim_{x \to 3} [x(x+1)]
(iii).  \lim_{x \to -1} [1+x+x^2+…+x^{10}]

Example 2. 
(i). \lim_{x \to 1} \left [ \frac{x^2+1}{x+100} \right ]
(ii). \lim_{x \to 2} \left [ \frac{x^3-4x^2+4x}{x^2-4} \right ]
(iii).  \lim_{x \to 2} \left [ \frac{x^2-4}{x^3-4x^2+4x} \right ]
(iv). \lim_{x \to 2} \left [ \frac{x^3-2x^2}{x^2-5x+6} \right ]
(v). \lim_{x \to 2} \left [ \frac{x-2}{x^2-x} – \frac{1}{x^3-3x^2+2x} \right ]

Limits and Derivatives Lecture 2
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Part - 2

NCERT EXERCISE 13.1

Basic identities for limits and method to use them

Question 13. \lim_{x \to 0} \frac{\sin ax}{bx}

Question 14. \lim_{x \to 0} \frac{\sin ax}{\sin bx}

Question 16. \lim_{x \to 0} \frac{\cos x}{\pi – x}

Question 17. \lim_{x \to 0} \frac{\cos 2x – 1}{\cos x-1}

Question 18. \lim_{x \to 0} \frac{ax + x \cos x}{b \sin x}

Question 20. \lim_{x \to 0} \frac{\sin ax + bx}{ax+ \sin bx} a, b, a+b \ne 0

Question 21. \lim_{x \to 0} (\cosec x – \cot x)