To find the point(s) of discontinuity for the function

$f(x) = \frac{x^2 - 4}{x^2 + 2x}$

we first need to identify any values of $x$ that make the denominator equal to zero, as these will potentially be points of discontinuity.

Step 1: Factor the numerator and denominator

The numerator is:

$x^2 - 4 = (x - 2)(x + 2)$

The denominator is:

$x^2 + 2x = x(x + 2)$

So the function becomes:

$f(x) = \frac{(x - 2)(x + 2)}{x(x + 2)}$

Step 2: Identify potential points of discontinuity

We can see that the denominator equals zero when:

$x = 0 \quad \text{or} \quad x = -2$

Now, let’s analyze each of these points.

• At $x = 0$, the denominator is zero, and the numerator is not zero. So, we have a potential infinite discontinuity.

• At $x = -2$, both the numerator and denominator are zero. This means we can simplify the function and possibly remove the discontinuity at this point. Let’s cancel the common factor of $(x + 2)$:

$f(x) = \frac{x - 2}{x}$

After canceling, the function becomes:

$f(x) = \frac{x - 2}{x}, \quad x \neq -2$

Since we removed the factor $(x + 2)$, the discontinuity at $x = -2$ is removable.

Step 3: Classify the discontinuities

• At $x = 0$, the denominator is zero but the numerator is not. Hence, this is an infinite discontinuity.

• At $x = -2$, the function can be simplified, so the discontinuity is removable.

Final Answer:

• Points of discontinuity: $x = 0, -2$

• Type of discontinuity at $x = -2$: removable

• Type of discontinuity at $x = 0$: infinite

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Would you like further details or clarification on any part of the solution? Here are some follow-up questions:

1. How can we identify the type of discontinuity in rational functions?
2. What happens to the function $f(x) = \frac{x-2}{x}$ at $x = 0$?
3. What is the difference between a jump discontinuity and an infinite discontinuity?
4. How can you simplify rational expressions to identify removable discontinuities?
5. Can there be more than one removable discontinuity in a rational function?

Tip: Always simplify rational functions when possible to see if discontinuities are removable, especially when both the numerator and denominator have common factors.