Evaluate the limit: lim(x→3) (x^2 - 9)/(x - 3).
Solution
Problem
Evaluate the limit: lim(x→3) (x^2 - 9)/(x - 3).
Step 1: Try direct substitution.
Substitute x = 3 into the expression:
(3^2 - 9)/(3 - 3) = 0/0
The result 0/0 is an indeterminate form, so we need to simplify the expression before evaluating the limit.
Step 2: Factor the numerator.
The numerator is a difference of squares:
x^2 - 9 = (x - 3)(x + 3)
So the expression becomes:
((x - 3)(x + 3))/(x - 3)
Step 3: Simplify.
For x ≠ 3, cancel the common factor (x - 3):
(x + 3)
Step 4: Evaluate the simplified limit.
Now substitute x = 3:
3 + 3 = 6
Therefore, the limit is 6.
Result
6