Evaluate the limit:
lim 25-(x+2)^2 / x-3
x->3
25(x-3) = 25 x -75
so
[ 25 x -75 - x^2 - 4 x - 4 ] /(x-3)
[-x^2 + 21 x -79 ] / (x-3)
(x-3) is not a factor of the numerator so I can not get rid of it in the denominator.
as x goes to 3 the numerator becomes -25 and the denominato is 0 so undefined is the answer.
If you mistyped it and mean
[ 25-(x+2)^2 ]/ (x-3 )
then
[ 25 - x^2 - 4 x - 4 ]/ (x-3)
that is
[ - x^2 - 4 x + 21 ] / (x-3)
- [ x^2 + 4 x - 21 ] / (x-3)
- (x-3)(x+7) / (x-3)
-x - 7
-3 - 7
-10
PLEASE US PARENTHESES SO WE CAN TELL NUMERATOR FROM DENOMINATOR !!!!!
sorry! the question was
lim [25 - (x+2)^2] / [x-3]
x->3
so
-10
I figured you must have mistyped it because (x-3) simply had to be a factor of the numerator or the question would not have been asked.
To evaluate the limit as x approaches 3, we can use direct substitution. Let's substitute x = 3 into the given expression:
lim [ 25 - (x+2)^2 ] / (x-3)
x->3
Plugging in x = 3:
[ 25 - (3+2)^2 ] / (3-3)
Simplifying further:
[ 25 - (5)^2 ] / 0
[ 25 - 25 ] / 0
0 / 0
We have obtained an indeterminate form (0/0) which means we can't determine the limit by direct substitution. We need to use a different method to evaluate this limit.
One possible approach is to factorize the numerator and simplify the expression. Let's factorize the numerator:
[ 25 - (x+2)^2 ] = [ 25 - (x^2 + 4x + 4) ]
= [ 25 - x^2 - 4x - 4 ]
= [ 21 - x^2 - 4x ]
Now, let's rewrite the expression:
lim [ 21 - x^2 - 4x ] / (x-3)
x->3
To simplify further, we can factor out a common factor of -1:
lim [ - (x^2 + 4x - 21) ] / (x-3)
x->3
Factoring the quadratic expression in the numerator, we get:
lim [ - (x + 7)(x - 3) ] / (x - 3)
x->3
Canceling out the common factor of (x - 3) in the numerator and denominator:
lim - (x + 7)
x->3
Substituting x = 3, we get:
- (3 + 7) = -10
Therefore, the limit of the given expression as x approaches 3 is -10.