Final answer:
The limit of the function (x² + x - 6) / (x + 3) as x approaches 3 is found by factoring the numerator, canceling out common factors, and then substituting x with 3, which results in the limit being equal to 1.
Step-by-step explanation:
To find the limit of the rational function (x² + x - 6) / (x + 3) as x approaches 3, we can first attempt to simplify the expression. If the denominator does not equal zero when x is substituted with 3, we could simply substitute the value of x. However, in this case, we encounter a division by zero which suggests a removable discontinuity or a hole in the graph at x = 3. To tackle this, we can factor the numerator to see if any factors cancel out with the denominator. The numerator x² + x - 6 factors to (x + 3)(x - 2).
After factoring, the function simplifies to:
(x - 2) for all x except x = -3
Now, as x approaches 3, the expression reduces to 3 - 2, which equals 1. Therefore, the limit as x approaches 3 is 1.