## Limits and Derivatives Lecture 4 Class 11

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