Math Snap

STEP 1

What is this asking?
We need to find the limit of a rational function as $x$ approaches negative infinity.
Watch out!
Don't forget to consider the degrees of the polynomials in the numerator and denominator!

STEP 2

1. Analyze the degrees
2. Divide by the highest power of $x$
3. Evaluate the limit

STEP 3

Alright, let's look at the degrees of the polynomials!
We've got a cubic polynomial $3x^3 - 2x^2 + 1$ in the numerator and a quadratic polynomial $x^2 + 1$ in the denominator.
Since the degree of the numerator is greater than the degree of the denominator, we expect the limit as $x$ approaches negative infinity to be either positive or negative infinity.

STEP 4

The highest power of $x$ in the entire expression is $x^3$.
Let's divide both the numerator and the denominator by $x^3$ to simplify things!
This is a clever trick because it helps us see what happens when $x$ gets really big (or really small, in this case!).

STEP 5

$$\lim_{x \to -\infty} \frac{3x^3 - 2x^2 + 1}{x^2 + 1} = \lim_{x \to -\infty} \frac{\frac{3x^3}{x^3} - \frac{2x^2}{x^3} + \frac{1}{x^3}}{\frac{x^2}{x^3} + \frac{1}{x^3}}$$

STEP 6

Now, let's simplify that expression!
$$\lim_{x \to -\infty} \frac{3 - \frac{2}{x} + \frac{1}{x^3}}{\frac{1}{x} + \frac{1}{x^3}}$$

STEP 7

As $x$ approaches negative infinity, terms like $\frac{2}{x}$, $\frac{1}{x^3}$, and $\frac{1}{x}$ all approach zero.
Think about it: as the denominator gets super huge (or super small, since it's negative), the whole fraction gets tiny!

STEP 8

So, we can rewrite our limit as:
$$\lim_{x \to -\infty} \frac{3 - \frac{2}{x} + \frac{1}{x^3}}{\frac{1}{x} + \frac{1}{x^3}} = \frac{3 - 0 + 0}{0 + 0} = \frac{3}{0}$$

STEP 9

Now, we've got a 3 in the numerator and a 0 in the denominator.
This means our limit is unbounded!
But we need to figure out if it's going to positive or negative infinity.

STEP 10

Since $x$ is approaching negative infinity, the terms $\frac{1}{x}$ and $\frac{1}{x^3}$ will be small negative numbers.
So, the denominator is approaching zero from the negative side.
Since the numerator is positive (3) and the denominator is approaching zero from the negative side, the whole fraction is going towards negative infinity!

The limit of the function as $x$ approaches negative infinity is $-\infty$.