Question

why doesnt the limit of 2x-12/|x+6| exist as you approach -6

Answer 1

because if x=-6 and u input that in the absolute value ur gonna get 0. when u muiltiply the rest by 0 u get 0

Answer 2

i guess i shouldnt have asked it like that because i understand that consept... my prof gave a hint and it says to evaluate the limit from the left and right

Answer 3

Split the absolute value sign into two parts.

Answer 4

\[
\frac{2x-12}{x+6}\quad \frac{2x-12}{-(x+6)}
\]

Answer 5

Suppose \[
\lim \frac{2x-12}{x+6} = L
\]Then \[
\lim \frac{2x-12}{-(x+6)} = -L
\]

Answer 6

The only way for a limit to exist is if \(L = -L\).

Answer 7

Or, basically it must be the case that \(L=0\).

Answer 8

We can solve the limit using l'Hospital's rule: \[
\lim \frac{2x-12}{x+6} = \lim \frac{(2x-12)'}{(x+6)'}= \lim \frac{2}{1} = 2
\]

Answer 9

Since \(L=2\neq 0\) the limit doesn't exist.