Evaluate the following limit: `underset( x rarr 0) ( "lim")(sqrt(2-x)-sqrt(2+x))/x`

Explanation

Given that limit: `lim_(x->0)(sqrt(2-x)-sqrt(2+x))/x`
`l=(sqrt(2-0)-sqrt(2+0))/0`
`=(0/0)`
As it has come as `0/0` form ,so we use L’Hospitals Rule.
now by using L’Hospitals Rule the limit is given by,
`l=lim_(x->0)(-1/(2(sqrt(2-x)))-1/(2(sqrt(2+x))))/1`
`=-1/(2sqrt(2))-1/(2sqrt(2))`
`=-1/sqrt(2)`