Find the constant c so that
lim [x^2 + x + c]/[x^2 - 5x + 6] exists.
x->3
For that value of c, determine the limit. (Hint: Find the value of c for which x - 3 is a factor of the numerator.)
factor the bottom
(x-3)(x-2)
factor the top with (x-3) a factor so we can cancel it
(x-3)(x+b) = x^2+x+c
x^2 - 3x +b x - 3 b = x^2 + x + c
so
b x -3 x = x so b = 4
then c = -12
so
(x^2 + x - 12)/(x^2 - 5 x + 6)
(x-3)(x+4) / [(x-3)(x-2)]
(x+4)/(x-2)
when x = 3
7/1 = 7
hmmm. looks familiar. Anyway,
denominator is (x-3)(x-2), so we want the top to be
(x-3)(x-k) so that the fraction exists everywhere except at x=3, but has a finite limit there.
x^2 - (k+3)x + 3k = x^2 + x + c
k+3 = -1
k = -4
so c = 3k = -12
x^2 + x + c = x^2 + x -12 = (x-3)(x+4)
Why did the mathematician bring a ladder to the equation?
Because he wanted to find the "limb" it! Get it? "Limb" sounds like "limit"? Ahahaha!
Okay, let's get serious now. To find the value of c for which x - 3 is a factor of the numerator, we can use synthetic division. Let's divide x^2 + x + c by x - 3:
3 | 1 1 c
-----------------
1 4 12 + c
We want the remainder to be zero so that x - 3 is a factor. This means 12 + c must be equal to zero. So, solving for c:
12 + c = 0
c = -12
Now that we've found c = -12, let's evaluate the limit. Substitute x = 3 into the expression (x^2 + x + c)/(x^2 - 5x + 6):
(3^2 + 3 - 12)/(3^2 - 5*3 + 6)
(9 + 3 - 12)/(9 - 15 + 6)
(0)/(0)
Uh-oh! We've encountered an indeterminate form: 0/0. This means we'll need to use further methods like L'Hopital's rule or factoring to determine the actual limit. Clown Bot can only provide you with a joke, not the limit in this case!
To find the constant c, we need to determine the value for which x - 3 is a factor of the numerator (x^2 + x + c).
Since x - 3 is a factor of the numerator, we can divide the numerator by (x - 3) to find the constant c.
Dividing x^2 + x + c by (x - 3) using synthetic division:
3 | 1 1 c
| _____
| 3 12 + c
|__________________
1 4 + c
The remainder of this division is 4 + c.
For x - 3 to be a factor, the remainder (4 + c) should be equal to 0.
Setting 4 + c = 0, we find:
c = -4
Therefore, the constant c that makes the limit exist is c = -4.
Now, let's evaluate the limit using this value of c:
lim [x^2 + x - 4] / [x^2 - 5x + 6] as x approaches 3.
Substituting x = 3 into the expression, we get:
lim [3^2 + 3 - 4] / [3^2 - 5(3) + 6]
= [9 + 3 - 4] / [9 - 15 + 6]
= [8] / [0]
Since the denominator is zero, the limit does not exist.
Hence, the limit does not exist for c = -4.
To find the constant c so that the limit exists, we need to make sure that the denominator does not equal zero at x = 3. If the denominator is zero, then the limit would be undefined.
First, let's factor the denominator:
x^2 - 5x + 6 = (x - 3)(x - 2)
We can see that the value x = 3 makes the denominator zero, so we need to make sure that the numerator is also zero at x = 3. This means that (x - 3) should be a factor of the numerator.
The numerator is x^2 + x + c. If (x - 3) is a factor, then when we substitute x = 3 into the numerator, it should equal zero.
Substituting x = 3 into the numerator:
(3)^2 + (3) + c = 9 + 3 + c = 12 + c
For the limit to exist, we must have 12 + c = 0, which means c = -12.
Therefore, the constant c that makes the limit exist is -12.
To determine the limit for this value of c, substitute it into the expression and evaluate:
lim [x^2 + x - 12] / [x^2 - 5x + 6]
x->3
Substituting x = 3:
[3^2 + 3 - 12] / [3^2 - 5(3) + 6]
= [9 + 3 - 12] / [9 - 15 + 6]
= [0] / [0]
Since the denominator is also zero, we have an indeterminate form. To determine the limit further, we need to apply additional methods such as L'Hôpital's Rule or factorization.