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.