Limits and Derivatives — Class 11 solved problems

32 problems from this chapter, each solved step by step.

Evaluate the limit as x approaches 0 of (e^(x²) - cos(x)) / (x²).
(3)/(2)
Evaluate the limit as n approaches infinity of sum from k=1 to n of n / (n² + k²).
(π)/(4)
Evaluate: lim(x->0) (sin(5x) - sin(3x)) / (sin(x)).
2
Find the derivative of the function given by f(x)=sin (x²).
f'(x) = 2x cos(x²)
Compute the derivative of 6 x^(100)-x^(55)+x.
600 x^(99) - 55 x^(54) + 1
lim_x → 0 csc x (√(2 cos² x + 3 cos x) - √(cos² x + sin x + 4)) is:
-(1)/(2√5)
Find the derivative of f(x) = x³ - 3x² + 2x - 5 with respect to x.
f'(x) = 3x² - 6x + 2
Evaluate: (i) lim _x → 1 (x^(15)-1)/(x^(10)-1) (ii) lim _x → 0 (√(1+x)-1)/(x)
(i) (3)/(2) (ii) (1)/(2)
The value of lim_n → ∞ ( Σ_k=1^n (k³ + 6k² + 11k + 6)/((k+3)!)) is:
e-1
If lim_t → 0 (∫_0^1 (3x + 5)^t dx)^((1)/(t)) = (2)/(56) ((8)/(5))^((2)/(α)), then α is equal to:
The problem statement contains an inconsistency. The calculated limit involves the constant e, which is not present in the given expression (2)/(56) ((8)/(5))^(
If Σ_r=1^n T_r = ((2n-1)(2n+1)(2n+3)(2n+5))/(64), then lim_n → ∞ Σ_r=1^n ((1)/(T_r)) is equal to:
(2)/(3)
Evaluate the limit: lim_x → ∞ (tan(5x^(1/3)) log_e(1 + 3x²))/((tan^(-1)(3√x))² (e^(5x^(4/3)) - 1)) is equal to:
The limit does not exist.
Find the derivative of the function f(x)=2 x²+3 x-5 at x=-1. Also prove that f^()(0)+3 f^()(-1)=0.
The derivative of the function f(x)=2 x²+3 x-5 at x=-1 is -1. The relation f^()(0)+3 f^()(-1)=0 is proven.
Find (d y)/(d x) if x-y=π.
(dy)/(dx) = 1
Compute the derivative of tan x.
The derivative of tan x is sec² x.
Prove that every polynomial of odd degree with real coefficients has at least one real root.
Every polynomial of odd degree with real coefficients has at least one real root.
Find the derivative of sin x at x=0.
The derivative of sin x at x=0 is 1.
Evaluate lim(n→∞) Σ(k=1 to n) [√(n² + k²) - n].
(1)/(6)
Find the derivative of f(x)=1+x+x²+x³+ +x^(50) at x=1.
The derivative of f(x) at x=1 is 1275.
If _x 0 ((tan x)/(x))^1/x²=p, then 96log_e p is equal to:
32
Let [t] be the greatest integer less than or equal to t. Then the least value of p N for which lim_x → 0^+ ( x ( [(1)/(x)] + [(2)/(x)] + + [(p)/(x)]) - x² ( [(1
No such natural number p exists.
If lim_x → ∞ ( ( (e)/(1-e)) ( (1)/(e) - (x)/(1+e)))^x = α, then the value of (log_e α)/(1 + log_e α) equals:
(1)/(2-e²)
Find the derivative of the constant function f(x)=a for a fixed real number a.
The derivative of the constant function f(x)=a is f'(x)=0.
Given below are two statements: **Statement I:** lim_x → 0 ( (tan^(-1)x + log_e ( (√(1 + x) - √(1 - x))/(x)) - 2x)/(x^5)) = (2)/(5) **Statement II:** lim_x → 1
Both Statement I and Statement II are false.
If lim_x 1 ((x-1)(6 + cos(x-1)) + sin(1-x))/((x-1)³) = -1 where, R then + is equal to:
18
For t > -1, let _t and _t be the roots of the equation ((t + 2)^((1)/(7)) - 1)x² + ((t + 2)^((1)/(6)) - 1)x + ((t + 2)^((1)/(21)) - 1) = 0. If lim_t → -1^+ _t =
98
Check the points where the constant function f(x)=k is continuous.
The constant function f(x)=k is continuous at all points x R.
If lim _x → 0 (cos(2x) + acos(4x) - b)/(x^4) is finite, then (a + b) is equal to:
(1)/(2)
Find the limits: (i) lim _x → 1[x³-x²+1] (ii) lim _x → 3[x(x+1)] (iii) lim _x →-1[1+x+x²+ +x^(10)].
(i) 1, (ii) 12, (iii) 1
Find the limits: (i) lim _x → 1[(x²+1)/(x+100)] (ii) lim _x → 2[(x³-4 x²+4 x)/(x²-4)] (iii) lim _x → 2[(x²-4)/(x³-4 x²+4 x)] (iv) lim _x → 2[(x³-2 x²)/(x²-5 x+6
(i) (2)/(101) (ii) 0 (iii) Does not exist (iv) -4 (v) 2
Find the derivative of f(x)=10 x.
The derivative of f(x)=10x is f'(x)=10.
Find the derivative of f(x)=x².
The derivative of f(x)=x² is f'(x)=2x.

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