# "One way this limit can be done is by rationalizing the" #
# "numerator here. We have:" #
# \qquad lim_{x rarr 1} { 1 - sqrt{ 2 - x } } / { x - 1 } \ = \ lim_{x rarr 1} { 1 - sqrt{ 2 - x } } / { x - 1 } cdot { 1 + sqrt{ 2 - x } } / { 1 + sqrt{ 2 - x } } #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ = \ lim_{x rarr 1} { 1 - sqrt{ 2 - x } } / { x - 1 } cdot { 1 + sqrt{ 2 - x } } / { 1 + sqrt{ 2 - x } } #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ = \ lim_{x rarr 1} { ( 1 - sqrt{ 2 - x } ) (1 + sqrt{ 2 - x } ) } / { ( x - 1 ) ( 1 + sqrt{ 2 - x } ) } #
# \qquad "continuing, using the difference of two squares:" #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ = \ lim_{x rarr 1} { ( 1 )^2 - sqrt{ 2 - x }^2 } / { ( x - 1 ) ( 1 + sqrt{ 2 - x } ) } #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ = \ lim_{x rarr 1} { 1 - ( 2 - x ) } / { ( x - 1 ) ( 1 + sqrt{ 2 - x } ) } #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ = \ lim_{x rarr 1} { 1 - 2 + x } / { ( x - 1 ) ( 1 + sqrt{ 2 - x } ) } #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ = \ lim_{x rarr 1} { -1 + x } / { ( x - 1 ) ( 1 + sqrt{ 2 - x } ) } #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ = \ lim_{x rarr 1} color{red}cancel{ x - 1 } / { color{red}cancel{ ( x - 1 ) }( 1 + sqrt{ 2 - x } ) } #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ = \ lim_{x rarr 1} 1 / { ( 1 + sqrt{ 2 - x } ) } #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ = \ 1 / { ( 1 + sqrt{ 2 - 1 } ) } #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ = \ 1 / { ( 1 + sqrt{ 1 } ) } #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ = \ 1 / { 1 + 1 } #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ = \ 1 / { 2 } \quad. #
# "This is our answer !" #
# "Summarizing, we have:" #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad lim_{x rarr 1} { 1 - sqrt{ 2 - x } } / { x - 1 } \ = \ 1 / { 2 } \quad. #
