Problem on elementary calculus

Question

Assignment:

Q1.  Let f: Z→ Z , where Z  is the set of integers and  f(x) = x⁵ + 101
     Is f(x) a one-to-one function?

Q2.  Prove that f(x) =4x - 3  is one-to-one function

Q3.  The ceiling function maps every real number to the smallest integer greater than or equal to that number, f : x  [x] , where [x] is the smallest integer greater than or equal to x . Is this function one-to-one? List five numbers that have the same image

Q4.  Find lim n³ + n / n² - 1  when n

Q5.  Find lim 3x⁷ + x³ +17 / 2x⁷ + x² +1 , as x

Q6.  Find lim x²⁰⁰⁸ + 1 / x²⁰⁰⁹ + 1 , as x

Q7.  Find lim x⁴ + 111,111 / x³ +111, 111, 111, as x

Q8. Prove that  f(x) = x² - 4  is a continuous function                 
    
Q9. Find derivatives of the following functions using differentiation rules

1.   f(x) = 9x +5

2.   f(x) = 3e^x

3.  f(x) =3x³ + 4

4.   f(x) = x⁴ +3x² +4x - 31

5.  f(x) = (2x +3)e^x 

6.  f(x) = (2x - 1)e^x-3                      

7.  f(x) = x(x - 2)²  = x(x² - 4x +4) = x³ - 4x²+ 4x

Q10. Find the derivative of the following function at  x_o = -1   
 
 f(x) = 3x² + x +1
  
Q11. Find differentials of the following functions:
1.  f(x) = 3x² + x+ 1                       
2.   f(x) = e^x-1                          
3.   f(x) = -e^x + x³ - 4                      

Q12. Let  f(x) = x  and  g(x) = x², where x ∈ R
Find f o g  and g o f as well as the domain and range of these functions.
Q13. Prove using the definition that lim 1/ n² = 0  , when  n→ ∞
Q14. Find the derivative of f(x) = 1/x by using the definition of the derivative.

Provide complete and step by step solution for the question and show calculations and use formulas.