Evaluate the following limit: `underset( x rarr 1) ( "lim")(sqrt(3+x)-sqrt(5-x))/(x^2-1)`

Explanation

`(lim)_(x->1)(sqrt(3+x)-sqrt(5-x))/(x^2-1)`


L'Hopital Rule:


`lim_(x rightarrow b) (f(x))/(g(x))=lim_(x rightarrow b) (f'(x))/(g'(x))`


`(lim)_(x->1)(1/sqrt(3+x)+1/sqrt(5-x))/(4x)`


`=(1/sqrt(3+1)+1/sqrt(5-1))/(4*1)`


`=(1/sqrt(4)+1/sqrt(4))/(4)`


`=(1/2+1/2)/(4)`


`=(1)/(4)`