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

Question

Isabella Ackerman · Accepted Answer

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#

Emily Ammerman · Answer

undefined To determine the limit of -4/(x-5) as x approaches 5, we can substitute the value of 5 into the expression and evaluate it. However, this would result in division by zero, which is undefined. Therefore, we need to use algebraic manipulation to simplify the expression. By factoring the denominator, we can rewrite the expression as -4/[(x-5)(-1)]. Canceling out the common factor of -1, we get 4/(5-x). Now, as x approaches 5, the expression 5-x approaches 0. Therefore, the limit of -4/(x-5) as x approaches 5 is -4/0, which is undefined.