#lim_(x->1) ((1-sqrt(2-x))/(x-1))#?

1 Answer
martin23239
Feb 21, 2018

# \qquad \qquad \qquad \qquad \qquad \qquad lim_{x rarr 1} { 1 - sqrt{ 2 - x } } / { x - 1 } \ = \ 1 / { 2 } \quad. #

Explanation:

# "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. #

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