How do you determine the limit of #-4/(x-5)# as x approaches 5?

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Answer 1 · 2016-07-05T11:58:17

left sided limit is #+oo#

right sided limit is #-oo#

well you can plug 5 in to find that the function amounts to #- 4/0# so there is a vertical asymptote there

the nature of the limit requires a little bit more thought

lets say you set #x = 5 + h, 0 < |h| "<<" 1# so we have

#lim_{h to 0} (- 4)/(5 + h -5)#

#=lim_{h to 0} -4/h#

for #h < 0#, ie left sided limit, the overall expression is positive so the left sided limit is #+ oo#

for #h > 0#, ie right sided limit, this is negative so the right sided limit is #- oo#