Why is it impossible to have #lim_(x-0) f(x)# and #lim_(f(x)-0)f(x)# simultaneously exist for any of these graphs?

Question

#A)# #f(x) = 1/x^2#
#B)# #f(x) = -1/x^2#
#C)# #f(x) = 1/x#
#D)# #f(x) = -1/x#

Christopher Allers · Accepted Answer

All four of these graphs have the #x#-axis as a horizontal asymptote as #x-> pm infty# and the #y#-axis as a vertical asymptote as #x->0# from the right or left.

Isaiah Allen · Answer

Well, by definition, a vertical asymptote is when at #x -> a#, #y -> pmoo# from either side and #x# never touches #a#. Similarly, a horizontal asymptote is when at #x -> pmoo#, #y -> b# from either side without ever reaching #b#.

For the function

#y = c/x#,

where #c# is a constant, if #x->0#, #y -> pmoo# from either side of #x = 0#, so you have a vertical asymptote. You can also find that as #x -> pmoo#, #y -> 0# but doesn't get there.

But if you have #x -> 0# and consequently #y -> pmoo#, you can't also have #x -> pmoo# so that #y -> 0#. It's not possible to approach both asymptotes at once because #x# cannot approach #0# and #pmoo# at the same time, and #y# cannot approach #pmoo# and #0# at the same time.

(Imagine trying to run to two different places at once; can't do it.)

Both kinds of asymptotes are on the graph, to be sure, but you can only approach one of those kinds of asymptotes at a time.

Christopher Agresta · Answer

undefined It is impossible to have both lim_(x->0) f(x) and lim_(f(x)->0) f(x) simultaneously exist for any of these graphs because the limit as x approaches 0 and the limit as f(x) approaches 0 are two different concepts. The limit as x approaches 0 refers to the behavior of the function as x gets arbitrarily close to 0, while the limit as f(x) approaches 0 refers to the behavior of the function as the output values approach 0. These two limits are not necessarily related and can have different outcomes.

Why is it impossible to have #lim_(x-&gt;0) f(x)# and #lim_(f(x)-&gt;0)f(x)# simultaneously exist for any of these graphs?

#A)# #f(x) = 1/x^2# #B)# #f(x) = -1/x^2# #C)# #f(x) = 1/x# #D)# #f(x) = -1/x#

Why is it impossible to have #lim_(x->0) f(x)# and #lim_(f(x)->0)f(x)# simultaneously exist for any of these graphs?

#A)# #f(x) = 1/x^2#
#B)# #f(x) = -1/x^2#
#C)# #f(x) = 1/x#
#D)# #f(x) = -1/x#