Help with limit

Question

lim x→3 (x3 − 13x2 + 51x − 63) / (x3 − 4x2 − 3x + 18) = a / 5

Find the value of a. (It is supposue to be -4. Why?)

Answer 1

Plug in x=3 will make the denominator equal to zero.  So we try to factor the numerator and denominator into a product of linear and quadratic factors.

 

Use synthetic division to factor.

 

For numerator:

 

3 | 1        -13         51         -63

 

                 3         -30          63

____________________________

      1       -10         21            0

 

 

(x - 3)(x² - 10x + 21) = (x - 3)(x - 3)(x - 7)

 

 

For denominator:

 

3 | 1      -4       -3         18

 

              3       -3        -18

__________________________

     1      -1       -6          0

 

 

(x - 3)(x² - x - 6) = (x - 3)(x - 3)(x - 2)

 

 

Then the limit becomes

 

lim          (x - 7) / (x - 2)

x-->3

 

 

Plug in x=3 and set the equation.

 

(3 - 7) / (3 - 2) = a / 5

 

-4 = a / 5

 

-20 = a

Answer 2

If you haven't gotten to L'Hopital's Rule yet, the numerator and denominator can be factored and simplified:

 

     lim x→3 (x-3)²(x-7)/(x-3)²(x+2) = lim x→3 (x-7)/(x+2) = (3-7)/(3+2) = -4/5

Answer 3

Since you have an indeterminate form, evaluate the limit using L'Hopital's rule.  Are you familiar with it?

 

lim x→3 of [(x³ − 13x² + 51x − 63) / (x³ − 4x² − 3x + 18)]

 

= lim x→3 of [(3x² - 26x + 51) / (3x² - 8x - 3)]

 

This form is still indeterminate, so apply L'Hopital's rule again.

 

lim x→3 of [(3x² - 26x + 51) / (3x² - 8x - 3)] = lim x→3 of [(6x - 26) / (6x - 8)] = -8/10 = -4/5

 

Then

 

a/5 = -4/5

 

a = -4