# How do you prove the statement lim as x approaches -3 for #(x^2+3x)# using the epsilon and delta definition?

See the explanation.

Initial Work

Proof

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To prove the statement lim as x approaches -3 for (x^2+3x), we need to show that for any given epsilon greater than zero, there exists a delta greater than zero such that if the distance between x and -3 is less than delta, then the distance between (x^2+3x) and the limit L is less than epsilon.

Let's proceed with the proof:

Given the function f(x) = x^2 + 3x, we want to find the limit as x approaches -3.

We start by assuming an arbitrary epsilon > 0.

We need to find a delta > 0 such that if |x - (-3)| < delta, then |(x^2 + 3x) - L| < epsilon.

Let's simplify the expression |(x^2 + 3x) - L|:

|(x^2 + 3x) - L| = |x^2 + 3x + 3|.

Now, we can manipulate the expression |x^2 + 3x + 3| to find a suitable delta.

We can rewrite |x^2 + 3x + 3| as |(x + 3)(x)|.

Since we are interested in the behavior of the function as x approaches -3, we can assume that |x + 3| < 1 (or any other positive value) to simplify the expression further.

Therefore, |(x + 3)(x)| < |1 * x| = |x|.

Now, we want to find a delta such that if |x - (-3)| < delta, then |x| < epsilon.

Since |x| < epsilon, we can choose delta = epsilon.

Thus, for any given epsilon > 0, if |x - (-3)| < delta = epsilon, then |(x^2 + 3x) - L| < epsilon.

Therefore, we have proven the statement using the epsilon and delta definition.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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