Question

find the limit as x approaches -6 of the function (2x+12)/|x+6| if it exists. If the limit does not exist, explain why.

Answer 1

lim = 0/0
differentiate top and bottom
derivative of top = 2
since bottom is absolute value it cannot be differentiated at x=-6 so we split it into 2 cases where x<-6 and x>-6
x<-6: derivative = -1
x>-6: derivative= 1
\[\lim_{x \rightarrow -6^{-}} = \frac{2}{-1}=-2\]
\[\lim_{x \rightarrow -6^{+}} = \frac{2}{1}= 2\]
Because the 2 limits do not agree the lim x->-6 does not exist