Chapter 13 Limits and derivatives

 

Q1. The value of the limit Lim_x→0 (cos x)^{cot2 x} is
(a) 1
(b) e
(c) e^1/2
(d) e^-1/2

Q2. The value of limit Lim_x→0 {sin (a + x) – sin (a – x)}/x is
(a) 0
(b) 1
(c) 2 cos a
(d) 2 sin a

Q3. Lim _x→-1 [1 + x + x² + ……….+ x¹⁰] is
(a) 0
(b) 1
(c) -1
(d) 2

Q4. The value of Lim_x→01 (1/x) × sin^-1 {2x/(1 + x²) is

(a) 0
(b) 1
(c) 2
(d) -2

Q5. Lim_x→0 log(1 – x) is equals to
(a) 0
(b) 1
(c) 1/2
(d) None of these

Q6. Lim_x→0 {(a^x – b^x)/ x} is equal to
(a) log a
(b) log b
(c) log (a/b)
(d) log (a×b)

Q7. The value of lim_y→0 {(x + y) × sec (x + y) – x × sec x}/y is
(a) x × tan x × sec x
(b) x × tan x × sec x + x × sec x
(c) tan x × sec x + sec x
(d) x × tan x × sec x + sec x

Q8. Lim_y→∞ {(x + 6)/(x + 1)}^(x+4) equals
(a) e
(b) e³
(c) e⁵
(d) e⁶

Q9. The derivative of [1+(1/x)] /[1-(1/x)] is
(a) 1/(x-1)²
(b) -1/(x-1)²
(c) 2/(x-1)²
(d) -2/(x-1)²

Q10. The expansion of log(1 – x) is
(a) x – x²/2 + x³/3 – ……..
(b) x + x²/2 + x³/3 + ……..
(c) -x + x²/2 – x³/3 + ……..
(d) -x – x²/2 – x³/3 – ……..

Q11. If f(x) = x × sin(1/x), x ≠ 0, then Lim_x→0 f(x) is
(a) 1
(b) 0
(c) -1
(d) does not exist

Q12. The value of Lim_n→∞ {1² + 2² + 3² + …… + n²}/n³ is
(a) 0
(b) 1
(c) -1
(d) n

Q13. The value of Lim_n→∞ (sin x/x) is
(a) 0
(b) 1
(c) -1
(d) None of these

Q14. The value of Lim_x→0 ax is
(a) 0
(b) 1
(c) 1/2
(d) 3/2

Q15. Let f(x) = cos x, when x ≥ 0 and f(x) = x + k, when x < 0 Find the value of k given that Lim_x→0 f(x) exists.
(a) 0
(b) 1
(c) -1
(d) None of these

Q16. The value of Lim_x→0 (1/x) × sin^-1 {2x/(1 + x²) is
(a) 0
(b) 1
(c) 2
(d) -2

Q17. Lim_x→0 sin (ax)/bx is
(a) 0
(b) 1
(c) a/b
(d) b/a

Q18. The value of the limit Lim_x→0 {log(1 + ax)}/x is
(a) 0
(b) 1
(c) a
(d) 1/a

Q19. If f(x) = (x + 1)/x then df(x)/dx is
(a) 1/x
(b) -1/x
(c) -1/x²
(d) 1/x²

Q20. Lim_x→0 (e^x² – cos x)/x² is equals to
(a) 0
(b) 1
(c) 2/3
(d) 3/2

Answer key

Q’s	1	2	3	4	5	6	7	8	9	10
Ans	d	c	b	c	a	c	d	c	d	d
Q’s	11	12	13	14	15	16	17	18	19	20
Ans	b	a	a	b	b	c	c	c	c	d