Exercise 12.1

Maximise Z = 3x + 4y subject to the constraints : x + y ≤ 4, x ≥ 0, y ≥ 0.

Maximise Z = x + y, subject to x - y ≤ -1, - x + y ≤ 0, x, y ≥ 0.

1/3 11/6 1lim z z z → − −

2/21 lim , 0 x

cx b x a→ + + + + ≠ + + 12. 1 1 2lim 2x x x→− + + 13. sinlim x ax bx→ 14. sinlim , , 0sinx ax a bbx→ ≠ 15. ( ) ( )π sin πlim π πx x x→ − − 16. coslim πx x x→ − 17. cos 2 1lim cos 1x x x→ − − 18. coslim sinx

b x→ + 19. 0 lim sec x x x → 20. sinlim , , 0sinx ax bx a b a bax bx→ + + ≠+ , 21. 0 lim (cosec cot ) x x x → − 22. π tan 2lim π x x x→ − 23. Find ( ) lim x f x → and ( ) lim x f x → , where ( ) ( ) 2 3, 0 3 1 , 0 x x

• ≤=  + > 24. Find ( ) lim x f x → , where ( ) 2/2 1, 1 1, 1 x xf x x x  − ≤= − − > 25. Evaluate ( ) lim x f x → , where ( ) | |, 0 0, 0 | x xf | x | x | |---|---|---| x  ≠=   = 26. Find ( ) lim x f x → , where ( ) , 0| | 0, 0 x xxf x x  ≠=   = 27. Find ( ) lim x f x → , where ( ) | | 5f x x = − 28. Suppose ( ) , 1 4, 1 , 1 a bx x | f | x | x | |---|---|---|

b ax x + < = =  − > and if lim x→ f (x) = f (1) what are possible values of a and b? 29. Let a1, a2, . . ., an be fixed real numbers and define a function ( ) ( ) ( ) ( )1 2 ... nf x x a x a x a= − − − . What is lim x a→ f (x) ? For some a ≠ a 1, a2, ..., a n, compute lim x a→ f (x). 30. If ( ) 1, 0 0, 0 1, 0 x x

x x /  + < = =  − > . For what value (s) of a does lim x a→ f (x) exists? 31. If the function f(x) satisfies ( ) 2lim π 1x f x x→ − = − , evaluate ( ) lim x f x → . 32. If ( ) 2/3 , 0 , 0 1 , 1

 + < = + ≤ ≤  + > . For what integers m and n does both ( ) lim x f x → and ( ) lim x f x → exist?

Minimise Z = - 3 x + 4 y subject to x + 2y ≤ 8, 3x + 2y ≤ 12, x ≥ 0, y ≥ 0.

Maximise Z = 5x + 3y subject to 3x + 5y ≤ 15, 5x + 2y ≤ 10, x ≥ 0, y ≥ 0.

Minimise Z = 3x + 5y such that x + 3y ≥ 3, x + y ≥ 2, x, y ≥ 0.

Maximise Z = 3x + 2y subject to x + 2y ≤ 10, 3 x + y ≤ 15, x, y ≥ 0.

Minimise Z = x + 2y subject to 2x + y ≥ 3, x + 2y ≥ 6, x, y ≥ 0. Show that the minimum of Z occurs at more than two points.

Minimise and Maximise Z = 5x + 10 y subject to x + 2y ≤ 120, x + y ≥ 60, x - 2y ≥ 0, x, y ≥ 0.

2/22 3 10lim 4x x x x→ − − − 8. 4/23 81lim 2 5 3x x x x→ − − − 9. lim 1x ax b cx→ + +

Minimise and Maximise Z = x + 2y subject to x + 2y ≥ 100, 2x - y ≤ 0, 2x + y ≤ 200; x, y ≥ 0.

Maximise Z = - x + 2y, subject to the constraints: x ≥ 3, x + y ≥ 5, x + 2y ≥ 6, y ≥ 0.