How do you determine the limit of #sqrt( x^2 + x) / (-2x)# as n approaches #-oo#?
1 Answer
The limit is
Explanation:
Here are the crucial observations:
(The above is true provided that
Also observe/recall that for real
Prepared with these two observations, we proceed:
# = lim_(xrarr-oo)((-x)sqrt(1+1/x))/(-2x)# #" "# [we have#x < 0# ]
# = lim_(xrarr-oo)(sqrt(1+1/x))/(2)#
# = sqrt(1+0)/2 = 1/2#
Bonus
Here is the limit as
# = lim_(xrarroo)((x)sqrt(1+1/x))/(-2x)# #" "# [we have#x > 0# ]
# = lim_(xrarroo)(sqrt(1+1/x))/(-2)#
# = sqrt(1+0)/(-2) = - 1/2#