Class 11 Limits and Derivatives Lecture 2

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When you can’t change the direction of the wind, just adjust your sails. – H. Jackson Brown Jr

Lecture - 2 Chapter 13 Limits and Derivatives

Booklets/Notes/Assignments are typed here on this website and their PDFs will be made available soon.

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

 

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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). \( \lim_{x \to 2} \left [ \frac{x-2}{x^2-x} – \frac{1}{x^3-3x^2+2x} \right ] \)

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

 

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