How do you determine the limit of 1/(x²+5x-6) as x approaches -6?

2 Answers
Apr 21, 2016

DNE-does not exist

Explanation:

lim_(x->-6) 1/((x+6)(x-1))

=1/(0*-7)

=1/0

DNE

Apr 21, 2016

The limit does not exist. Look at the signs of the factors.

Explanation:

Let f(x) = 1/(x^2+5x-6) = 1/((x+6)(x-1))

Not that as xrarr-6, we have (x-1) rarr -7

From the left

As xrarr-6^-, the factor (x+6)rarr0^-, so f(x) is positive and increasing without bound.

lim_(xrarr-6^-)f(x) = oo

From the right

As xrarr-6^+, the factor (x+6)rarr0^+, so f(x) is negative and increasing without bound.

lim_(xrarr-6^+)f(x) = -oo

Two sided

lim_(xrarr-6)f(x) does not exist.