Calculus 1 Weiqi Zhou
we call the limit of as , if for any , there exists some , so that holds for all in
is the limit of as ，if converges to for any with (for all ) that converges to
to have a limit at , need not be defined at but shall be defined near
show that
for any , take
if ，then holds for all in
if ，then also holds for all in
show that does not have a limit as
take any positive sequence that converges to ，then converges to as it is a constant sequence
take any negative sequence that converges to ，then converges to
the limit as approach from one side (i.e., from left or from right)
is the left limit of as , if for any , there exists some , so that holds for all in
is the left limit of as ，if converges to for any with (for all ) that converges to
is the right limit of as , if for any , there exists some , so that holds for all in
is the right limit of as ，if converges to for any with (for all ) that converges to
: is the left limit of as
: is the right limit of as
compute the left limit and the right limit of
as respectively
is the limit of as , if for any , there exists some , so that holds for all in
it is also viable to define as
Show that