Limits and Derivatives Lecture 4 Class 11

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We become what we think about most of the time, and that’s the strangest secret. – Earl Nightingale

Lecture - 4 Chapter 13 Limits and Derivatives

Finding limits in double definition functions

NCERT EXERCISE 13.1

Question 23. Find \( \lim_{x \to 0} f(x)\) and \( \lim_{x \to 1} f(x) \) , where \( f(x) = \begin{cases} {2x+3}, & x \le 0 \\ 3(x+1), & x>0 \end{cases} \)

Question 24. Find \( \lim_{x \to 1} f(x) \) , where \( f(x) = \begin{cases} {x^2-1}, & x \le 1 \\ {-x^2-1}, & x>0 \end{cases} \)

Finding limits in double definition functions

Question 25. Find \( \lim_{x \to 0} f(x) \) , where \( f(x) = \begin{cases} {\frac{|x|}{x}}, & x \ne 0 \\ {0}, & x=0 \end{cases} \)

Question 26. Find \( \lim_{x \to 0} f(x) \) , where \( f(x) = \begin{cases} {\frac{x}{|x|}}, & x \ne 0 \\ {0}, & x=0 \end{cases} \)

Question 27. Find \( \lim_{x \to 5} f(x) \) , where \( f(x) = |x|-5 \)

Question 28. Suppose \( f(x) = \begin{cases} {a+bx}, & x<1 \\ {4}, & x=0  \\ {b-ax}, & x>1 \end{cases} \) and if \( \lim_{x \to 1} f(x) = f(1)\) what are the possible values of a and b?

Question 29. Let \(a_1, a_2, \dots, a_n\) be fixed real numbers and define a function \(f(x) = (x-a_1)(x-a_2) \dots (x-a_n)\). What is \( \lim_{x \to a_1} f(x)\) ? For some \(a \ne a_1, a_2, \dots,  a_n\), compute \( \lim_{x \to a} f(x)\).

Question 30. If \( f(x) = \begin{cases} {|x|+1}, & x<0 \\ {0}, & x=0 \\ {|x|-1}, & x>0 \end{cases} \). For what value (s) of a does \( \lim_{x \to a} f(x)\) exists?

Question 31. If the function \( f(x) \) satisfies \( \lim_{x \to 1} \frac{f(x)-2}{x^2-1} = \pi \), evaluate \( \lim_{x \to 1} f(x)\).

Question 32. If \( f(x) = \begin{cases} {mx^2+n}, & x<0 \\ {nx+m}, & 0 \le x \le 1 \\ {nx^3+m}, & x>1 \end{cases} \)

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