Find the value of a at limit x=2.

[TsAmE]

Homework Statement

Is there a number a such that

lim
x -> 2 3x^(2) + ax + a + 3 / x^(2) + x - 2

exists? If so, find the value of a and the value of the limit.

Homework Equations

None.

The Attempt at a Solution

lim
x -> 2 (x + 2)(3x + p) / (x + 2)(x-1)

(factored top containing x + 2, as it has to cancel with the bottem for limit to exist and made 3x + p = rest of polynomial)

Middle term: 3x 2
X
x p

3xp + 2x = ax
3p + 2 = a

constant:

2p = a + 3
2p = 3p + 2 + 3
p = -5

3(-5) + 2 = a
a = -13

but the answer at the back of my book for a was 15

 

[HallsofIvy]

Homework Statement

Is there a number a such that

lim
x -> 2 3x^(2) + ax + a + 3 / x^(2) + x - 2

exists? If so, find the value of a and the value of the limit.

Are you sure the limit is at 2 and not -2? At x= 2, the denominator is 4+ 2- 2= 4, not 0 so this limit exists and is (15+ 3a)/4 for any value of a.

If the limit is as x goes to -2, then the denominator is 0 and the limit will exist if and only if the numerator is also 0. $3(-2)^2+ a(-2)+ a+ 3= -a+ 15$ so a must be 15.

Homework Equations

None.

The Attempt at a Solution

lim
x -> 2 (x + 2)(3x + p) / (x + 2)(x-1)

(factored top containing x + 2, as it has to cancel with the bottem for limit to exist and made 3x + p = rest of polynomial)

Middle term: 3x 2
X
x p

3xp + 2x = ax
3p + 2 = a

constant:

2p = a + 3
2p = 3p + 2 + 3
p = -5

3(-5) + 2 = a
a = -13

but the answer at the back of my book for a was 15