Question

using definition of limit.. prove Lt x->-1 (x+5)/(2x+3) is 4 ??

Answer 1

                       x + 5
Lim        f (x) = -------      =  4
x-> -1              2x+3
using the definition.
you have to prove that, given an epsilon  ∈ , there exists a delta d such that
| f(x) - 4 | < ∈    whenever  | x - (-1) |  < d  that is  | x + 1 | < d.

So  let   f(x) - 4 < ∈      = >  f(x) < 4 + ∈
(x+5)/(2x+3)  < 4+∈    =>  x+5  <  (8+2∈)x +12 + 3∈
=> x (-7-2∈) < 7  + 3∈
=> x < - (7 +3∈) / (7+2∈)
=> x+1 < 1  - (7 +3∈) / (7+2∈)
=> x+1  <  -∈/7+2∈   this is equal to d.
since there exists a 'd' for every ∈. Limit exists and it is  4.

Answer 2

(x+5)(2x+)=4........3x+3x+10x + 15=4......x=-11/16.