Question 1: 

lim_x->3 (x + 3)

lim_{x-> π} (x – )

lim_{r-> 1} (πr²)

lim_{x-> 4} [(4x + 3) ÷ (x - 2)]

lim_{x-> -1} (x¹⁰ + x⁵ + 1) ÷ (x - 1)

lim_{x-> 0} {(x + 1)⁵ - 1)} ÷ x

lim_{x-> 2} {(3x² – x - 10) ÷ (x² - 4)} 

lim_{x-> 3} {(x⁴ – 81) ÷ (2x² – 5x - 3)} 

lim_{x-> 0} {(ax + b) ÷ (cx + 1)}

lim_{z-> 1} {(z^{1 ÷ 3} - 1) ÷ (z^{1 ÷ 6} - 1)}

lim_{x-> 1} {(ax² + bx + c) ÷ (cx² + bx + a)}, a + b + c ≠ 0

lim_{x-> 1} {(  + ) ÷ (x + 2)}

lim_{x-> -0} (sin )

lim_{x-> -0} ( ), a, b ≠ 0

lim_{x-> π} [sin (π - x) ÷ {π (π - x)}]

lim_{x-> 0} [cos x ÷ (π - x)]

lim_{x-> 0} [(cos 2x – 1) ÷ (cos x – 1)]

lim_{x-> 0} [(ax + x × cos x) ÷ (b × sin x)]

lim_{x-> 0} [x × sec x]

lim_{x-> 0} [(sin ax + bx) ÷ (ax + sin bx)], a, b, a + b ≠ 0

lim_{x-> 0} (cosec x – cot x)

lim_x->   [tan 2x ÷ (x – )]

Find lim_{x-> 0} f(x) and lim_{x-> 1} f(x)

where f(x) = 2x + 3, {x ≤ 0 3(x + 1), {x > 0   }

Find

lim_{x-> 1} f(x) where f(x) = x² - 1, {x ≤ 1 = -x² - 1, {x > 1 }

Find lim_{x-> 0} f(x)

where f(x) = |x| ÷ x, {x ≠ 0

= 0, {x = 0  

Find lim_{x-> 0} f(x) where f(x) = x ÷|x|, {x ≠ 0 = 0, {x = 0  

Find lim_{x-> 5} f(x) where f(x) = |x| - 5

Suppose                       

a + bx, {if x < 1                     

f(x) = 4, {if x = 0  b – ax, {if x > 1

and lim_x->1 f(x) = f (1). What are the possible values of a and b?

Let a₁, a₂, a₃, ………………, a_n be fixed real numbers and define a function f(x) = (x – a₁) (x – a₂) (x – a₃) …………… (x – a_n).

What is lim_{x – a1} f(x)? For some a ≠ a₁, a₂, a₃, ………………,a_n , compute lim_{x – a} f(x).

If f(x) =|x| + 1, {if x < 0

If f(x) = 0, {if x = 0   

If f(x) = |x| - 1, {if x > 0

For what value(s) of a does lim_x->a f(x) exists?

If the function f(x) satisfies lim _{x ->1} [f(x) – 2] ÷ [x² - 1] = π, evaluate lim _{x ->1} f(x).

mx² + n, {if x<0 {If f(x) = nx + m, {if 0 < x ≤ 1  nx³ + m, {if x >1

For what integers m and n does lim_x->0 f(x) and lim_x->1 f(x) exists?

Find the derivative of x² – 2 at x = 10.

Find the derivative of 99x at x = l00.

Find the derivative of the following functions from first principle.

(1) x³ – 27

(2) (x – 1) (x – 2)

(3) 1 ÷ x² 

(4) (x + 1) ÷ (x - 1)       

For the function f(x) = (x¹⁰⁰ ÷ 100) + (x⁹⁹ ÷ 99) + ……... + (x² ÷ 2) + x + 1.

Prove that f’ (1) = 100f’ (0).

Find the derivative of xⁿ + ax^n-1 + a² x^n-2 + ………+ a^n-1 x + aⁿ for some fixed real number a.

For some constants a and b, find the derivative of

(1) (x – a) (x – b)

(2) (ax² + b)²                         

(3) (x - a) ÷ (x - b)

Find the derivative of {(xⁿ - aⁿ) ÷ (x - a)} for some constant a.

Find the derivative of

(1) 2x –                       

(2) (5x³ + 3x – 1) (x – 1)    

(3) x^–3(5 + 3x)                                              

(4) x⁵(3 – 6x^–9)                   

(5) x^–4 (3 – 4x^–5)                                    

(6) {2 ÷ (x + 1)} – {x² ÷ (3x - 1)}

Find the derivative of cos x from first principle.

Find the derivative of the following functions:

1. sin x cos x

2. sec x                            

3. 5 sec x + 4 cos x              

4. cosec x 

5. 3cot x + 5cosec x    

6. 5sin x – 6cos x + 7    

7. 2tan x – 7sec x

Find the derivative of the following functions from first principles:

1. −x

2. (-x)^-1     

3. sin (x + 1)                    

4. cos (x – )

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are

fixed non-zero constants and m and n are integers): (x + a)

(px + q) (  + s)

(ax + b) (cx + d)²

(ax + b) ÷ (cx + d)

(1 + ) ÷ (1 – )

1 ÷ (ax² + bx + c)

(ax + b) ÷ (px² + qx + r) 

(px² + qx + r) ÷ (ax + b)

(a ÷ x⁴) – (b ÷ x²) + cos x

4√x - 2

(ax + b) ⁿ

(ax + b) ⁿ (cx + d) ^m  

sin (x + a)

cosec x × cot x

(sin x + cos x) ÷ (sin x - cos x)

(Sec x - 1) ÷ (sec x + 1)

sinⁿ x

(a + b × sin x) ÷ (c + d × cos x)

x⁴(5 sin x – 3 cos x)

(x² + 1) cos x

(ax² + sin x) (p + q cos x)

(x + cos x) (x – tan x)

(4x + 5 sin x) ÷ (3x + 7 cos x)

{x² cos ( )} ÷ sin x

x ÷ (1 + tan x)

(x + sec x) (x – tan x)

x ÷ sinⁿ x