Question

Find all values of
`c` such that the limit exists.


`image`
(Give your answer in the form of a comma-separated list. Express numbers in exact form. Use symbolic notation and fractions where needed. Enter DNE if there are no values of
`c` such that the limit exists.)

Answer 1

For
`\lim_(x\rightarrow1)(x^2+6x+c)/(x-1)` to exist, (x-1) must be a factor of the numerator.

If x-1 is a factor of the numerator, then x=1 is a root of the numerator.

This means
`(1)^2+6(1)+c=0`.

Then 1+6+c=0, or 7+c=0, or c = –7.

`\lim_(x\rightarrow1)(x^2+6x-7)/(x-1) = \lim_(x\rightarrow1)((x-1)(x+7))/(x-1) =\lim_(x\rightarrow1)(x+7) = 8`

c = –7

For any other value of c, you'll be left with x-1 in the denominator and you won't have a limit that exists.