Find a delta for any epsilon given: x^2+x-6/x+3 and x0=-3
To find a delta for any epsilon for the given expression, we need to use the definition of a limit. Let's walk through the steps.
Step 1: Write down the expression and the given value:
The expression is f(x) = (x^2 + x - 6) / (x + 3). And x0 = -3.
Step 2: Express the limit definition:
We have to find a delta > 0 such that if 0 < |x - (-3)| < delta, then |f(x) - L| < epsilon. Here, L is the limit value we wish to find, and epsilon is the tolerance or error we allow.
Step 3: Calculate the limit value:
To find the limit, we substitute x = -3 into the expression:
f(-3) = (-3^2 + (-3) - 6) / (-3 + 3)
= (9 - 3 - 6) / 0
= 0 / 0 (indeterminate form)
The limit here is indeterminate, which means we need further steps. We can use algebraic manipulation or other techniques such as factoring or canceling common factors to simplify the expression before proceeding.
Step 4: Simplify the expression:
Factor the numerator:
f(x) = [(x - 2)(x + 3)] / (x + 3)
Cancel out the common factor:
f(x) = (x - 2)
We now have a simplified expression without any indeterminate forms.
Step 5: Express the inequality |f(x) - L| < epsilon:
|f(x) - L| < epsilon
|(x - 2) - L| < epsilon
Step 6: Solve for delta:
Since we want to find a delta for any epsilon, we need to find a relationship between delta and epsilon that will guarantee the inequality holds.
|(x - 2) - L| < epsilon can be rewritten as:
|x - 2| < epsilon + |L|
We can choose delta = epsilon + |L|, where |L| is a positive constant. This selection ensures that if 0 < |x - (-3)| < delta, then |f(x) - L| < epsilon.
Therefore, a delta for any epsilon would be delta = epsilon + |L|. In this case, L is 2.