Finding 6 In Exercises 43-48, find the limit L. Then find S such that |/(r) - L| < e whenever 0 < [r - c| < 8 for *)
€ = 0.01 and (b) € = 0.005
45. lim as x goes to 2 of (x^2-3)

Let's analyze the problem and find the limit \( L \) and \( \delta \) for the given function and conditions.

### Step 1: Find the Limit \( L \)

We are asked to find the limit of the function \( f(x) = x^2 - 3 \) as \( x \) approaches 2:

\[
\lim_{x \to 2} (x^2 - 3)
\]

To find this limit, we can substitute \( x = 2 \) into the function:

\[
f(2) = 2^2 - 3 = 4 - 3 = 1
\]

Thus, the limit \( L \) is 1:

\[
L = 1
\]

### Step 2: Find \( \delta \) for Given \( \epsilon \) Values

We need to find \( \delta \) such that:

\[
|f(x) - L| < \epsilon \quad \text{whenever} \quad 0 < |x - 2| < \delta
\]

#### (a) For \( \epsilon = 0.01 \)

Let's express the condition \( |f(x) - L| < \epsilon \):

\[
|x^2 - 3 - 1| < 0.01
\]

Simplify the expression inside the absolute value:

\[
|x^2 - 4| < 0.01
\]

We can factor this as:

\[
|(x - 2)(x + 2)| < 0.01
\]

To relate this to \( |x - 2| \), consider:

\[
|x - 2| \cdot |x + 2| < 0.01
\]

Now, assume \( \delta \) is small enough such that \( |x - 2| < 1 \). Then, \( 1 < x < 3 \), and consequently, \( 3 < x + 2 < 5 \). Thus, \( |x + 2| < 5 \).

Using this bound:

\[
|x - 2| \cdot 5 < 0.01
\]

\[
|x - 2| < \frac{0.01}{5} = 0.002
\]

Therefore, for \( \epsilon = 0.01 \), we can choose \( \delta = 0.002 \).

#### (b) For \( \epsilon = 0.005 \)

Repeat the process for \( \epsilon = 0.005 \):

\[
|x^2 - 4| < 0.005
\]

\[
|(x - 2)(x + 2)| < 0.005
\]

Again, using \( |x + 2| < 5 \):

\[
|x - 2| \cdot 5 < 0.005
\]

\[
|x - 2| < \frac{0.005}{5} = 0.001
\]

Thus, for \( \epsilon = 0.005 \), we can choose \( \delta = 0.001 \).

### Summary

1. **Limit \( L \)**: \( 1 \)
2. **\( \delta \) for \( \epsilon = 0.01 \)**: \( 0.002 \)
3. **\( \delta \) for \( \epsilon = 0.005 \)**: \( 0.001 \)

Would you like more details, or do you have any questions?

### Related Questions

1. How do we handle limits when the function has a discontinuity at the point of interest?
2. What is the formal definition of a limit using \( \epsilon \) and \( \delta \)?
3. How do we determine \( \delta \) when \( |x - c| < \delta \) for more complex functions?
4. How do limits relate to derivatives?
5. What happens to the limit if \( x \to \infty \) or \( x \to -\infty \)?

### Tip

When finding \( \delta \) for a given \( \epsilon \), it helps to estimate bounds on \( |x + c| \) or any expression to simplify the inequality.

Learn how to find the limit of the function (x^2 - 3) as x approaches 2. This problem involves using the epsilon-delta definition to determine the appropriate delta for given epsilon values (0.01 and 0.005). Detailed step-by-step solution included.

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