# \ #
# \mbox{I'm not completely sure of the expression given,} \ \ \mbox{but I suspect it is as below (please let me know if I am wrong !!)} #
# \mbox{We want:} #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad lim_{x rarr 0} { \sqrt{ 4 - x } - \sqrt{ 4 + x } } / x. #
# \mbox{One convenient way to do it (there are others) is to use} \ \ \mbox{the conjugate of the numerator of this expression, like below.} \ \mbox{(Don't let all the radicals bother you, if they did -- } \ \mbox{it's much easier than it may look !)} #
# lim_{x rarr 0} { \sqrt{ 4 - x } - \sqrt{ 4 + x } } / x \ = #
# \qquad \qquad \qquad lim_{x rarr 0} ( { \sqrt{ 4 - x } - \sqrt{ 4 + x } } / x \cdot { \sqrt{ 4 - x } + \sqrt{ 4 + x } } / { \sqrt{ 4 - x } + \sqrt{ 4 + x } } ) \ = #
# \qquad \qquad \qquad lim_{x rarr 0} ( ( \sqrt{ 4 - x } )^2 -( \sqrt{ 4 + x } )^2 } / { x \cdot ( \sqrt{ 4 - x } + \sqrt{ 4 + x } ) } \ = #
# \qquad \qquad \qquad lim_{x rarr 0} { ( 4 - x ) -( 4 + x ) } / { x \cdot ( \sqrt{ 4 - x } + \sqrt{ 4 + x } ) } \ = #
# \qquad \qquad \qquad lim_{x rarr 0} { 4 - x -4 - x } / { x \cdot ( \sqrt{ 4 - x } + \sqrt{ 4 + x } ) } \ = #
# \qquad \qquad \qquad lim_{x rarr 0} { - 2 x } / { x \cdot ( \sqrt{ 4 - x } + \sqrt{ 4 + x } ) } \ = #
# \qquad \qquad \qquad lim_{x rarr 0} { - 2 color(red){ cancel{x} } } / { color(red){ cancel{x} } \cdot ( \sqrt{ 4 - x } + \sqrt{ 4 + x } ) } \ = #
# \qquad \qquad \qquad lim_{x rarr 0} { - 2 } / { ( \sqrt{ 4 - x } + \sqrt{ 4 + x } ) } \ = #
# \qquad \qquad \qquad \qquad \qquad \qquad { - 2 } / { ( \sqrt{ 4 - 0 } + \sqrt{ 4 + 0 } ) } \ = #
# \qquad \qquad \qquad \qquad \qquad \qquad { - 2 } / { ( \sqrt{ 4 } + \sqrt{ 4 } ) } \ = \ { - 2 } / { 2 + 2 } \ = \ { - color(red){ cancel{2} } } / { 2 \cdot color(red){ cancel{2} } } \ = \ -1/2. #
# \ #
# \mbox{Final Statement:} #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad lim_{x rarr 0} { \sqrt{ 4 - x } - \sqrt{ 4 + x } } / x \ = \ -1/2. #
